data/EuclideaGame.csv

pack,level_id,question,solution_nl,solution_tool,solution_symbol,initial_symbol
Alpha,https://euclidea.fandom.com/wiki/Equilateral_Triangle,"Description: Construct an equilateral triangle with the given side. The construction must be exact. Guessing even very close points does not count as a solution.Let A and B be the two end points of the given segment.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center A and radius AB
<Circle Tool>Construct a circle with center B and radius AB, intersecting circle A at points C and D
<Line Tool>Construct line AC
<Line Tool>Construct line BC","['Circle Tool', 'Circle Tool', 'Line Tool', 'Line Tool']","[['A', 'B'], ['AB'], ['C', 'D'], ['AC'], ['BC']]","['A', 'B']"
Alpha,https://euclidea.fandom.com/wiki/Angle_of_60%C2%B0,"Description: Construct an angle of 60° with the given side.Let A be the initial point of the given ray.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center A and an arbitrary radius, intersecting the ray at point B.
<Circle Tool>Construct a circle with center B and radius AB, intersecting circle A at points C and D.
<Line Tool>Construct line AC.","['Circle Tool', 'Circle Tool', 'Line Tool']","[['A'], ['B'], ['C', 'AB', 'D'], ['AC']]",['A']
Alpha,https://euclidea.fandom.com/wiki/Perpendicular_Bisector,"Description: Construct the perpendicular bisector of the segment.Given the segment AB.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center A and radius AB.
<Circle Tool>Construct a circle with center B and radius AB, intersecting circle A at C and D.
<Line Tool>Construct line CD.","['Circle Tool', 'Circle Tool', 'Line Tool']","[['AB'], ['A'], ['C', 'B', 'D'], ['CD']]",['AB']
Alpha,https://euclidea.fandom.com/wiki/Midpoint,"Description: Construct the midpoint of the segment defined by two points.Let A and B be the two points.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB.
<Perpendicular Bisector Tool>Construct the perpendicular bisector of AB, intersecting AB at point C.","['Line Tool', 'Perpendicular Bisector Tool']","[['A', 'B'], ['AB'], ['C']]","['A', 'B']"
Alpha,https://euclidea.fandom.com/wiki/Circle_in_Square,"Description: Inscribe a circle in the square.Given the square ABCD.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC.
<Perpendicular Bisector Tool>Construct the perpendicular bisector of AB, intersecting AB at point E and AC at point F.
<Circle Tool>Construct a circle with center E and radius EF.","['Line Tool', 'Perpendicular Bisector Tool', 'Circle Tool']","[['ABCD'], ['AC'], ['F', 'AB', 'E'], ['EF']]",['ABCD']
Alpha,https://euclidea.fandom.com/wiki/Rhombus_in_Rectangle,"Description: Inscribe a rhombus in the rectangle so that they share a diagonal.Given the rectangle ABCD with AB>AD.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Bisector Tool>Construct the perpendicular bisector of AC, intersecting AB at point E and CD at point F.
<Line Tool>Construct line AF.
<Line Tool>Construct line CE.","['Perpendicular Bisector Tool', 'Line Tool', 'Line Tool']","[['ABCD', 'AB', 'AD'], ['F', 'E', 'AC', 'CD'], ['AF'], ['CE']]","['ABCD', 'AB', 'AD']"
Alpha,https://euclidea.fandom.com/wiki/Circle_Center,"Description: Construct the center of the circle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the perpendicular bisector P1 of two arbitrary points on the circle.
<Circle Tool>Construct the perpendicular bisector of two other arbitrary points on the circle, intersecting P1 at point A.
<Point Tool>Point at A.      ","['Circle Tool', 'Circle Tool', 'Point Tool']","[[], [], ['A'], []]",[]
Alpha,https://euclidea.fandom.com/wiki/Inscribed_Square,"Description: Inscribe a square in the circle. One vertex of the square is given.Let O be the center of the circle and A the vertex.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line OA, intersecting the circle at point B.
<Perpendicular Bisector Tool>Construct the perpendicular bisector of AB, intersecting the circle at points C and D.
<Line Tool>Construct line AC.
<Line Tool>Construct line BC.
<Line Tool>Construct line BD.
<Line Tool>Construct line AD.","['Line Tool', 'Perpendicular Bisector Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['O', 'A'], ['B', 'OA'], ['C', 'AB', 'D'], ['AC'], ['BC'], ['BD'], ['AD']]","['O', 'A']"
Beta,https://euclidea.fandom.com/wiki/Angle_Bisector,"Description: Construct the line that bisects the given angle.Let A be the vertex of the angle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center A and an arbitrary radius, intersecting the sides of the angle at points B and C.
<Perpendicular Bisector Tool>Construct the perpendicular bisector of BC.","['Circle Tool', 'Perpendicular Bisector Tool']","[['A'], ['B', 'C'], ['BC']]",['A']
Beta,https://euclidea.fandom.com/wiki/Intersection_of_Angle_Bisectors,"Description: Construct the point where the angle bisectors of the triangle are intersected.Given the triangle ABC.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Angle Bisector Tool>Construct the angle bisector A1 of  angle BAC.
<Angle Bisector Tool>Construct the angle bisector A2 of  angle ACB, intersecting A1 at D.","['Angle Bisector Tool', 'Angle Bisector Tool']","[['ABC'], ['BAC'], ['ACB', 'D']]",['ABC']
Beta,https://euclidea.fandom.com/wiki/Angle_of_30%C2%B0,"Description: Construct an angle of 30° with the given side.Let A be the initial point of the given ray.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with an arbitrary center B and radius AB, intersecting the ray at point C. Although the image shown here places B on the ray, this construction works even when point B is not on the ray, so long as B is on the same half of the plane as the ray.
<Circle Tool>Construct a circle with center C and radius BC, intersecting circle B at points D and E.
<Line Tool>Construct line AD.","['Circle Tool', 'Circle Tool', 'Line Tool']","[['A'], ['AB', 'C', 'B'], ['BC', 'E', 'D'], ['AD']]",['A']
Beta,https://euclidea.fandom.com/wiki/Double_Angle,"Description: Construct an angle equal to the given one so that they share one side.Let O be the vertex of the angle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with an arbitrary center A on one ray and an arbitrary radius so that it intersects the other ray at least once. Let B be one of those intersections.
<Circle Tool>Construct a circle with another arbitrary center C on the first ray and radius BC, intersecting circle A at D.
<Line Tool>Construct line OD.","['Circle Tool', 'Circle Tool', 'Line Tool']","[['O'], ['B', 'A'], ['C', 'BC', 'D'], ['OD']]",['O']
Beta,https://euclidea.fandom.com/wiki/Cut_Rectangle,"Description: Construct a line through the given point that cuts the rectangle into two parts of equal area.Given the rectangle ABCD and the point O.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC, a diagonal of the rectangle
<Line Tool>Construct line BD, the other diagonal, intersecting AC at E
<Line Tool>Construct line OE","['Line Tool', 'Line Tool', 'Line Tool']","[['ABCD', 'O'], ['AC'], ['BD', 'E'], ['OE']]","['ABCD', 'O']"
Beta,https://euclidea.fandom.com/wiki/Drop_a_Perpendicular,"Description: Drop a perpendicular from the point to the line.Let O be the point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a circle with center O and an arbitrary radius so that it intersects the line at two points A and B
<Perpendicular Bisector Tool>Construct the perpendicular bisector of AB","['Line Tool', 'Perpendicular Bisector Tool']","[['O'], ['B', 'A'], ['AB']]",['O']
Beta,https://euclidea.fandom.com/wiki/Erect_a_Perpendicular,"Description: Erect a perpendicular from the point on the line.Let O be the point, and A and B two arbitrary points on the line so that O is between them.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ",<Angle Bisector Tool>Construct the angle bisector of  angle AOB,['Angle Bisector Tool'],"[['O', 'A', 'B', 'O'], ['AOB']]","['O', 'A', 'B', 'O']"
Beta,https://euclidea.fandom.com/wiki/Perpendicular_Tool,"Description: You have discovered the Perpendicular tool. Click on the line, then on the point.Let A be a line and B be a point, whether B is on or off line A (the level shows B off of A).
Tool List: ['Perpendicular Tool']
Solution: ","<Perpendicular Tool>With the perpendicular tool active, select the line, then the point.",['Perpendicular Tool'],"[['A', 'B', 'B', 'A', 'B', 'A'], []]","['A', 'B', 'B', 'A', 'B', 'A']"
Beta,https://euclidea.fandom.com/wiki/Tangent_to_Circle_at_Point,"Description: Construct a tangent to the circle at the given point.Let O be the center of the circle and A the point on it.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line OA
<Perpendicular Tool>Construct the perpendicular to the line OA from point A","['Line Tool', 'Perpendicular Tool']","[['O', 'A'], ['OA'], []]","['O', 'A']"
Beta,https://euclidea.fandom.com/wiki/Circle_Tangent_to_Line,"Description: Construct a circle with the given center that is tangent to the given line.Let O be the center.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to the line from point O, intersecting the line at A
<Circle Tool>Construct the circle with center O and radius OA","['Perpendicular Tool', 'Circle Tool']","[['O'], ['A'], ['OA']]",['O']
Beta,https://euclidea.fandom.com/wiki/Circle_in_Rhombus,"Description: Inscribe a circle in the rhombus.Given the rhombus ABCD.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC, a diagonal of the rhombus
<Line Tool>Construct line BD, the other diagonal of the rhombus, intersecting AC at O
<Perpendicular Tool>Construct the perpendicular to AB through O, intersecting AB at E
<Circle Tool>Construct the circle with center O radius OE","['Line Tool', 'Line Tool', 'Perpendicular Tool', 'Circle Tool']","[['ABCD'], ['AC'], ['BD', 'O'], ['E', 'AB'], ['OE']]",['ABCD']
Gamma,https://euclidea.fandom.com/wiki/Chord_Midpoint,"Description: Construct a chord whose midpoint is given.Let A be the center and B the given midpoint.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB
<Perpendicular Tool>Construct the perpendicular to AB through B","['Line Tool', 'Perpendicular Tool']","[['A', 'B'], ['AB'], []]","['A', 'B']"
Gamma,https://euclidea.fandom.com/wiki/Triangle_by_Angle_and_Orthocenter,"Description: Construct a segment connecting the sides of the angle to get a triangle whose orthocenter is in the point O.Let A be the vertex of the angle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to one side R1 of the angle through O, intersecting the second side at A
<Perpendicular Tool>Construct the perpendicular to the second side R2 of the angle through O, intersecting the first side at B
<Line Tool>Construct line AB","['Perpendicular Tool', 'Perpendicular Tool', 'Line Tool']","[['O', 'A'], [], ['B'], ['AB']]","['O', 'A']"
Gamma,https://euclidea.fandom.com/wiki/Intersection_of_Perpendicular_Bisectors,"Description: Construct a segment connecting the sides of the angle to get a triangle whose perpendicular bisectors are intersected in the point O.Let A be the vertex of the angle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the circle with center O and radius OA, intersecting the sides of the angle at B and C
<Line Tool>Construct line BC","['Circle Tool', 'Line Tool']","[['O', 'A'], ['B', 'C', 'OA'], ['BC']]","['O', 'A']"
Gamma,https://euclidea.fandom.com/wiki/Three_equal_segments_-_1,"Description: Given an angle ABC and a point M inside it, find points D on BA and E on BC and construct segments DM and ME such that BD = DM = ME.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of BM, intersecting AB at D
<Circle Tool>Construct the circle with center M and radius DM, intersecting BC at E and F
<Line Tool>Construct line DM
<Line Tool>Construct line ME","['Perpendicular Tool', 'Circle Tool', 'Line Tool', 'Line Tool']","[['ABC', 'M', 'D', 'BA', 'E', 'BC', 'DM', 'ME', 'BD', 'DM', 'ME'], ['AB', 'BM'], ['F'], [], []]","['ABC', 'M', 'D', 'BA', 'E', 'BC', 'DM', 'ME', 'BD', 'DM', 'ME']"
Gamma,https://euclidea.fandom.com/wiki/Circle_through_Point_Tangent_to_Line,"Description: Construct a circle through the point A that is tangent to the given line at the point B.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB
<Perpendicular Tool>Construct the perpendicular to the line from B, intersecting the perpendicular bisector at O
<Circle Tool>Construct the circle with center O and radius OA (or OB)","['Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool']","[['A', 'B'], ['AB'], ['O'], ['OB', 'OA']]","['A', 'B']"
Gamma,https://euclidea.fandom.com/wiki/Midpoints_of_Trapezoid_Bases,"Description: Construct a line passing through the midpoints of the trapezoid bases.Given the trapezoid ABCD.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of the base AB, intersecting AB at E
<Perpendicular Tool>Construct the perpendicular bisector of the base CD, intersecting CD at F
<Line Tool>Construct line EF","['Perpendicular Tool', 'Perpendicular Tool', 'Line Tool']","[['ABCD'], ['AB', 'E'], ['F', 'CD'], ['EF']]",['ABCD']
Gamma,https://euclidea.fandom.com/wiki/Angle_of_45%C2%B0,"Description: Construct an angle of 45° with the given side.Let A be the initial point of the given ray and B a distinct arbitrary point on that ray.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to the ray from A; let C be a distinct arbitrary point on that perpendicular
<Angle Bisector Tool>Construct the angle bisector of  angle BAC","['Perpendicular Tool', 'Angle Bisector Tool']","[['A', 'B'], ['C'], ['BAC']]","['A', 'B']"
Gamma,https://euclidea.fandom.com/wiki/Lozenge,"Description: Construct a rhombus with the given side and an angle of 45° in a vertex.Given the side AB (A on the left to match the orientation in explore mode)
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to AB from A; let C be a distinct arbitrary point on that perpendicular (above A, to match the orientation in explore mode)
<Angle Bisector Tool>Construct the angle bisector of  angle BAC
<Circle Tool>Construct the circle with center A and radius AB, intersecting the angle bisector at D and line AC at E with E on the side of A opposite from C
<Perpendicular Tool>Construct the perpendicular to AC through D
<Line Tool>Construct line BE","['Perpendicular Tool', 'Angle Bisector Tool', 'Circle Tool', 'Perpendicular Tool', 'Line Tool']","[['AB', 'A'], ['C'], ['BAC'], ['AC', 'E', 'D'], [], ['BE']]","['AB', 'A']"
Gamma,https://euclidea.fandom.com/wiki/Center_of_Quadrilateral,"Description: Construct the midpoint of the segment that connects the midpoints of the diagonals of the quadrilateral.Given the quadrilateral ABCD (arbitrary which vertex is A)
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at E
<Perpendicular Tool>Construct the perpendicular bisector of CD, intersecting CD at F
<Perpendicular Tool>Construct the perpendicular bisector of EF
<Line Tool>Construct line EF
<Perpendicular Tool>Point at the intersection of EF and its perpendicular bisector","['Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Line Tool', 'Perpendicular Tool']","[['ABCD', 'A'], ['AB', 'E'], ['F', 'CD'], ['EF'], [], []]","['ABCD', 'A']"
Delta,https://euclidea.fandom.com/wiki/Double_Segment,"Description: Construct a point C on the line AB such that |AC|=2|AB| using only a compass.
Tool List: ['Point Tool', 'Circle Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the circle with center A and radius AB
<Circle Tool>Construct the circle with center B and radius AB, intersecting circle A at D and E
<Circle Tool>Construct the circle with center D and radius DE, intersecting circle B at C
<Point Tool>Point at C","['Circle Tool', 'Circle Tool', 'Circle Tool', 'Point Tool']","[['C', 'AB', 'AC', 'AB'], ['A'], ['E', 'D', 'B'], ['DE'], []]","['C', 'AB', 'AC', 'AB']"
Delta,https://euclidea.fandom.com/wiki/Angle_of_60%C2%B0_-_2,"Description: Construct a straight line through the given point that makes an angle of 60° with the given line.Let A be the given point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle B with radius AB, intersecting the line at point C
<Circle Tool>Construct circle C with radius AC, intersecting circle B at point D
<Circle Tool>Construct circle D with radius CD, intersecting circle C at points E and F (E on the left to match the orientation in explore mode)
<Line Tool>Construct line AE","['Circle Tool', 'Circle Tool', 'Circle Tool', 'Line Tool']","[['A'], ['AB', 'C', 'B'], ['D', 'AC'], ['F', 'E', 'CD'], ['AE']]",['A']
Delta,https://euclidea.fandom.com/wiki/Circumscribed_Equilateral_Triangle,"Description: Construct an equilateral triangle that is circumscribed about the circle and contains the given point.Let O be the center of the circle and A the point on it.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line OA, intersecting the circle at B
<Perpendicular Tool>Construct the perpendicular to OA from A
<Circle Tool>Construct the circle with center B and radius OB, intersecting the circle at C and D, and line OA at E
<Line Tool>Construct line CE
<Line Tool>Construct line DE","['Line Tool', 'Perpendicular Tool', 'Circle Tool', 'Line Tool', 'Line Tool']","[['O', 'A'], ['B', 'OA'], [], ['C', 'OB', 'E', 'D'], ['CE'], ['DE']]","['O', 'A']"
Delta,https://euclidea.fandom.com/wiki/Equilateral_Triangle_in_Circle,"Description: Inscribe an equilateral triangle in the circle using the given point as a vertex. The center of the circle is not given.Let A be the vertex.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center A and an arbitrary radius, intersecting the original circle at B and C
<Circle Tool>Construct the circle with center B and radius AB, intersecting circle A at D and E
<Perpendicular Tool>Construct the perpendicular bisector of CD, intersecting the original circle at F
<Perpendicular Tool>Construct the perpendicular bisector of CE, intersecting the original circle at G
<Line Tool>Construct line FG (total 9E)","['Circle Tool', 'Circle Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Line Tool']","[['A'], ['B', 'C'], ['AB', 'D', 'E'], ['F', 'CD'], ['G', 'CE'], ['FG']]",['A']
Delta,https://euclidea.fandom.com/wiki/Cut_Two_Rectangles,"Description: Construct a line that cuts each of the rectangles into two parts of equal area.Given the rectangles ABCD and EFGH.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC, a diagonal of the first rectangle
<Line Tool>Construct line BD, the other diagonal, intersecting the previous diagonal at I
<Line Tool>Construct line EG, a diagonal of the second rectangle
<Line Tool>Construct line FH, the other diagonal, intersecting the previous diagonal at J
<Line Tool>Construct line IJ","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['ABCD', 'EFGH'], ['AC'], ['BD', 'I'], ['EG'], ['FH', 'J'], ['IJ']]","['ABCD', 'EFGH']"
Delta,https://euclidea.fandom.com/wiki/Square_Root_of_2,"Description: Let |AB|=1. Construct a point C on the ray AB such that the length of AC is equal to √2.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to AB from B
<Circle Tool>Construct the circle with center B and radius AB, intersecting the perpendicular at D
<Circle Tool>Construct the circle with center A and radius AD, intersecting the ray at C
<Point Tool>Point at C","['Perpendicular Tool', 'Circle Tool', 'Circle Tool', 'Point Tool']","[['AB', 'C', 'AB', 'AC'], ['B'], ['D'], ['AD', 'A'], []]","['AB', 'C', 'AB', 'AC']"
Delta,https://euclidea.fandom.com/wiki/Square_Root_of_3,"Description: Let |AB|=1. Construct a point C on the ray AB such that the length of AC is equal to √3.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the circle with center B and radius AB, intersecting the ray at D
<Circle Tool>Construct the circle with center D and radius BD, intersecting circle B at E
<Circle Tool>Construct the circle with center A and radius AE, intersecting the ray at C
<Point Tool>Point at C","['Circle Tool', 'Circle Tool', 'Circle Tool', 'Point Tool']","[['AB', 'C', 'AB', 'AC'], ['B', 'D'], ['BD', 'E'], ['AE', 'A'], []]","['AB', 'C', 'AB', 'AC']"
Delta,https://euclidea.fandom.com/wiki/Angle_of_15%C2%B0,"Description: Construct an angle of 15° with the given side.Let A be the initial point of the given ray.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with an arbitrary center B and radius AB, intersecting the ray at C
<Circle Tool>Construct the circle with center C and radius BC, intersecting circle B at D and E (D above ray to match orientation of explore mode)
<Angle Bisector Tool>Construct the angle bisector of  angle DAC","['Circle Tool', 'Circle Tool', 'Angle Bisector Tool']","[['A'], ['AB', 'C', 'B'], ['BC', 'E', 'D'], ['DAC']]",['A']
Delta,https://euclidea.fandom.com/wiki/Square_by_Opposite_Midpoints,"Description: Construct a square, given two midpoints of opposite sides.Let A and B be the given midpoints.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the circle with center A and radius AB
<Circle Tool>Construct the circle with center B and radius AB, intersecting circle A at C and D.
<Circle Tool>Construct the circle with center C and radius CB
<Line Tool>We draw the line (AC) that crosses the circle of center C in D.","['Circle Tool', 'Circle Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['AB'], ['C', 'D'], ['CB'], ['AC']]","['A', 'B']"
Delta,https://euclidea.fandom.com/wiki/Square_by_Adjacent_Midpoints,"Description: Construct a square, given two midpoints of adjacent sides.Let A and B be the given midpoints.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB
<Perpendicular Tool>Construct the perpendicular to AB from A
<Perpendicular Tool>Construct the angle bisector of  angle BAC with C an arbitrary point on the perpendicular (to match the orientation of explore mode, choose C above A)
<Perpendicular Tool>Construct the perpendicular to that angle bisector through B, intersecting AC at D
<Circle Tool>Construct the circle with center B and radius BD, intersecting BD at E and AC at F
<Perpendicular Tool>Construct the perpendicular bisector of BE
<Perpendicular Tool>Construct the perpendicular to the angle bisector of  angle BAC through F","['Line Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool', 'Perpendicular Tool', 'Perpendicular Tool']","[['A', 'B'], ['AB'], [], ['BAC', 'C'], ['D', 'AC'], ['F', 'BD', 'E'], ['BE'], []]","['A', 'B']"
Epsilon,https://euclidea.fandom.com/wiki/Parallel_Line,"Description: Construct a line parallel to the given line and through the given point.Let A be the given point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to the line through A
<Perpendicular Tool>Construct the perpendicular to the previous perpendicular from A","['Perpendicular Tool', 'Perpendicular Tool']","[['A'], [], []]",['A']
Epsilon,https://euclidea.fandom.com/wiki/Parallel_Tool,"Description: You have discovered the Parallel tool. Click on the line, then on the point.
Tool List: ['Parallel Tool']
Solution: ","<Parallel Tool>With the parallel tool active, click on the line then on the point.",['Parallel Tool'],"[[], []]",[]
Epsilon,https://euclidea.fandom.com/wiki/Parallelogram_by_Three_Vertices,"Description: Construct a parallelogram whose three of four vertices are given.Let A, B and C be the given vertices (to match the orientation of explore mode, place A on top and B on the left)
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB
<Line Tool>Construct line BC
<Parallel Tool>Construct the parallel to AB through C
<Parallel Tool>Construct the parallel to BC through A","['Line Tool', 'Line Tool', 'Parallel Tool', 'Parallel Tool']","[['A', 'B', 'C', 'A', 'B'], ['AB'], ['BC'], [], []]","['A', 'B', 'C', 'A', 'B']"
Epsilon,https://euclidea.fandom.com/wiki/Line_Equidistant_from_Two_Points_-_1,"Description: Construct a line through the point C and at equal distance from the points A and B but that does not pass between them.Given points A, B, and C (C already labeled, choice of A and B is arbitrary):
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB
<Parallel Tool>Construct the parallel to AB through C","['Line Tool', 'Parallel Tool']","[['C', 'A', 'B', 'A', 'B', 'C', 'C', 'A', 'B'], ['AB'], []]","['C', 'A', 'B', 'A', 'B', 'C', 'C', 'A', 'B']"
Epsilon,https://euclidea.fandom.com/wiki/Line_Equidistant_from_Two_Points_-_2,"Description: Construct a line through the point C that goes between the points A and B and that is at equal distance from them.Given points A, B, and C, with C labeled and the choice of A and B arbitrary:
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB
<Line Tool>Construct line AB, intersecting its perpendicular bisector at D
<Line Tool>Construct line CD","['Perpendicular Tool', 'Line Tool', 'Line Tool']","[['C', 'A', 'B', 'A', 'B', 'C', 'C', 'A', 'B'], ['AB'], ['D'], ['CD']]","['C', 'A', 'B', 'A', 'B', 'C', 'C', 'A', 'B']"
Epsilon,https://euclidea.fandom.com/wiki/Hash,"Description: Construct a line through the given point on which two pairs of parallel lines cut off equal line segments.Let O be the given point, and ABCD the parallelogram created by the two pairs of parallel lines (A on top to match the orientation of explore mode):
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC, a diagonal of the parallelogram
<Parallel Tool>Construct the parallel to AC through O","['Line Tool', 'Parallel Tool']","[['O', 'ABCD', 'A'], ['AC'], []]","['O', 'ABCD', 'A']"
Epsilon,https://euclidea.fandom.com/wiki/Shift_Angle,"Description: Construct an angle from the given point that is equal to the given angle so that their sides are parallel.Let O be the given point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Angle Bisector Tool>Construct the parallel to one side of the angle through O
<Angle Bisector Tool>Construct the parallel to the other side of the angle through O","['Angle Bisector Tool', 'Angle Bisector Tool']","[['O'], [], []]",['O']
Epsilon,https://euclidea.fandom.com/wiki/Line_Equidistant_from_Two_Lines,"Description: Construct a straight line parallel to the given parallel lines that lies at equal distance from them.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to one line from an arbitrary point A on it, intersecting the other line at B
<Perpendicular Tool>Construct the perpendicular bisector of AB","['Perpendicular Tool', 'Perpendicular Tool']","[[], ['B', 'A'], ['AB']]",[]
Epsilon,https://euclidea.fandom.com/wiki/Circumscribed_Square,"Description: Circumscribe a square about the circle. Two of its sides should be parallel to the given line.Let O be the center of the given circle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to the line through O, intersecting the circle at A and B
<Perpendicular Tool>Construct the perpendicular to AB from A
<Perpendicular Tool>Construct the perpendicular to AB from B
<Circle Tool>Construct the circle with center A and radius OA, intersecting the perpendicular to AB from A at C and D
<Perpendicular Tool>Construct the perpendicular to CD from C
<Perpendicular Tool>Construct the perpendicular to CD from D (total 16E)","['Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool', 'Perpendicular Tool', 'Perpendicular Tool']","[['O'], ['B', 'A'], ['AB'], [], ['C', 'D', 'OA'], ['CD'], []]",['O']
Epsilon,https://euclidea.fandom.com/wiki/Square_in_Square,"Description: Inscribe a square in the square. A vertex is given.Given the square ABCD and E the given vertex.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the circle with center A and radius AE, intersecting AD at F
<Perpendicular Tool>Construct the perpendicular to AD through F, intersecting BC at G
<Line Tool>Construct line EG
<Perpendicular Tool>Construct the perpendicular to EG from E, intersecting AD at H
<Perpendicular Tool>Construct the perpendicular to EG from G, intersecting CD at I
<Line Tool>Construct the line HI (total 12E)","['Circle Tool', 'Perpendicular Tool', 'Line Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Line Tool']","[['A', 'ABCD', 'E'], ['F', 'AD', 'AE'], ['G', 'BC'], ['EG'], ['H'], ['I', 'CD'], ['HI']]","['A', 'ABCD', 'E']"
Epsilon,https://euclidea.fandom.com/wiki/Circle_Tangent_to_Square_Side,"Description: Construct a circle that is tangent to a side of the square and goes through the vertices of the opposite side.Given the square ABCD (AB on the bottom for the solution to match the hint mode orientation)
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting CD at E and AB at F
<Perpendicular Tool>Construct the perpendicular bisector of AE, intersecting EF at G
<Circle Tool>Construct the circle with center G and radius AG (or EG)","['Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool']","[['ABCD', 'AB'], ['F', 'E', 'CD'], ['EF', 'G', 'AE'], ['AG', 'EG']]","['ABCD', 'AB']"
Epsilon,https://euclidea.fandom.com/wiki/Regular_Hexagon,"Description: Construct a regular hexagon with the given side.Given the side AB.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct the circle with center A and radius AB
<Circle Tool>Construct the circle with center B and radius AB, intersecting circle A at C and D (C below AB to match the orientation of explore mode)
<Line Tool>Construct line AC, intersecting circle A at E
<Line Tool>Construct line BC, intersecting circle B at F
<Parallel Tool>Construct the parallel to AC through F
<Parallel Tool>Construct the parallel to BC through E, intersecting the the parallel to AC at G
<Perpendicular Tool>Construct the perpendicular bisector of DG (total 15E)","['Circle Tool', 'Circle Tool', 'Line Tool', 'Line Tool', 'Parallel Tool', 'Parallel Tool', 'Perpendicular Tool']","[['AB'], ['A'], ['C', 'B', 'D'], ['E', 'AC'], ['F', 'BC'], [], ['G'], ['DG']]",['AB']
Zeta,https://euclidea.fandom.com/wiki/Point_Reflection,"Description: Reflect the segment across the point.Given the segment AB and O the point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AO
<Circle Tool>Construct the circle with center O and radius AO, intersecting AO at C
<Parallel Tool>Construct the parallel to AB through C
<Line Tool>Construct line BO, intersecting the parallel to AB at D
<Point Tool>Point at D","['Line Tool', 'Circle Tool', 'Parallel Tool', 'Line Tool', 'Point Tool']","[['AB', 'O'], ['AO'], ['C'], [], ['BO', 'D'], []]","['AB', 'O']"
Zeta,https://euclidea.fandom.com/wiki/Reflection,"Description: Reflect the segment across the line.Given the segment AB.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a circle with an arbitrary center C on the line and radius AC
<Line Tool>Construct a circle with another arbitrary center D on the line and radius AD, intersecting circle C at E
<Line Tool>Construct a circle with an arbitrary center F on the line and radius BF
<Line Tool>Construct a circle with another arbitrary center G on the line and radius BG, intersecting circle F at H
<Line Tool>Construct line EH","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['AB'], ['C', 'AC'], ['AD', 'E', 'D'], ['F', 'BF'], ['H', 'G', 'BG'], ['EH']]",['AB']
Zeta,https://euclidea.fandom.com/wiki/Copy_Segment,"Description: Construct a segment from the given point that is equal to the given segment and lies on the same line with it.Given the segment AB and the point C.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC
<Perpendicular Tool>Construct the perpendicular bisector of AC, intersecting AC at D
<Circle Tool>Construct the circle with center D and radius BD, intersecting AC at E
<Point Tool>Point at E","['Line Tool', 'Perpendicular Tool', 'Circle Tool', 'Point Tool']","[['AB', 'C'], ['AC'], ['D'], ['BD', 'E'], []]","['AB', 'C']"
Zeta,https://euclidea.fandom.com/wiki/Given_Angle_Bisector,"Description: Construct two straight lines through the two given points respectively so that the given line is a bisector of the angle that they make.Let A and B be the given points.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a circle with an arbitrary center C on the line and radius AC
<Line Tool>Construct a circle with another arbitrary center D on the line and radius AD, intersecting circle C at E
<Line Tool>Construct line BE, intersecting the line at F
<Line Tool>Construct line AF","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['A', 'B'], ['C', 'AC'], ['AD', 'E', 'D'], ['BE', 'F'], ['AF']]","['A', 'B']"
Zeta,https://euclidea.fandom.com/wiki/Non-collapsing_Compass,"Description: Construct a circle with the given center and the radius equal to the length of the given segment.Given the segment AB and the point C.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of A and C.
<Perpendicular Tool>Construct the circle with center arbitrary point D on the perpendicular bisector of A and C and with radius DB.
<Perpendicular Tool>Construct the circle with center arbitrary point E on the perpendicular bisector of A and C and with radius EB, intersecting circle D at F.
<Circle Tool>Construct the circle with center C and radius CF. (6E)","['Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool']","[['AB', 'C'], ['A'], ['DB', 'D'], ['F', 'EB', 'E'], ['CF']]","['AB', 'C']"
Zeta,https://euclidea.fandom.com/wiki/Compass_Tool,"Description: You have discovered the Non-collapsing Compass tool. Click on two points to set the radius length, then on the center of the circle.
Tool List: ['Compass Tool']
Solution: ","<Compass Tool>With the compass tool active, click on one endpoint of the segment, then another endpoint of the segment, then on the destination point.",['Compass Tool'],"[[], []]",[]
Zeta,https://euclidea.fandom.com/wiki/Translate_Segment,"Description: Construct a segment from the given point parallel and equal to the given segment.Given the segment AB and the point C.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Use the non-collapsing compass to construct the circle with radius AB and center C
<Parallel Tool>Construct the parallel to AB through C, intersecting the circle at two points
<Line Tool>Use the intersect tool on the circle and parallel line to select both points at once","['Circle Tool', 'Parallel Tool', 'Line Tool']","[['AB', 'C'], [], [], []]","['AB', 'C']"
Zeta,https://euclidea.fandom.com/wiki/Triangle_by_Three_Sides,"Description: Construct a triangle with the side AB and the two other sides equal to the given segments.Given the other segments CD and EF.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Use the non-collapsing compass to construct the circle with radius CD and center A
<Circle Tool>Use the non-collapsing compass to construct the circle with radius EF and center B, intersecting circle A at G and H
<Line Tool>Construct line AG
<Line Tool>Construct line BG","['Circle Tool', 'Circle Tool', 'Line Tool', 'Line Tool']","[['AB', 'CD', 'EF'], ['A'], ['H', 'G', 'B'], ['AG'], ['BG']]","['AB', 'CD', 'EF']"
Zeta,https://euclidea.fandom.com/wiki/Parallelogram,"Description: Construct a parallelogram with the given side and the midpoint of the opposite side in the given point.Given the segment AB and the point C.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at D
<Circle Tool>Use the non-collapsing compass to construct the circle with radius AD and center C
<Parallel Tool>Construct the parallel to AB through C, intersecting circle C at E and F
<Line Tool>Construct line AE
<Line Tool>Construct line BF","['Perpendicular Tool', 'Circle Tool', 'Parallel Tool', 'Line Tool', 'Line Tool']","[['AB', 'C'], ['D'], ['AD'], ['F', 'E'], ['AE'], ['BF']]","['AB', 'C']"
Zeta,https://euclidea.fandom.com/wiki/Nine_Point_Circle,"Description: Construct a circle that passes through the midpoints of sides of the given acute triangle.Given the triangle ABC.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to AB through C, intersecting AB at D
<Perpendicular Tool>Construct the perpendicular to BC through A, intersecting BC at E
<Perpendicular Tool>Construct the perpendicular bisector of DE, intersecting AC at F
<Perpendicular Tool>Construct the perpendicular bisector of DF, intersecting the perpendicular bisector of DE at G
<Circle Tool>Construct the circle with center G and radius DG","['Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool']","[['ABC'], ['C', 'AB', 'D'], ['BC', 'E', 'A'], ['F', 'DE', 'AC'], ['G', 'DF'], ['DG']]",['ABC']
Zeta,https://euclidea.fandom.com/wiki/Symmetry_of_Four_Lines,"Description: Three lines are intersected in a point. Construct a line so that the set of all 4 lines is mirror symmetric.Let O be the intersection of the three lines.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center O and an arbitrary radius, intersecting the lines at six points; let A, B and C be a set of three consecutive points
<Circle Tool>Use the non-collapsing compass to construct the circle with radius AB and center C, intersecting circle O at D and E
<Line Tool>Construct line OD","['Circle Tool', 'Circle Tool', 'Line Tool']","[['O'], ['B', 'C', 'A'], ['E', 'D', 'AB'], ['OD']]",['O']
Zeta,https://euclidea.fandom.com/wiki/Parallelogram_by_Three_Midpoints,"Description: Construct a parallelogram given three of the midpoints.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Use the non-collapsing compass to construct the circle with radius AC and center B
<Circle Tool>Use the non-collapsing compass to construct the circle with radius BC and center A, intersecting circle B at D and E, with D on the opposite side of C from AB
<Line Tool>Construct line CD
<Parallel Tool>Construct the parallel to CD through A
<Parallel Tool>Construct the parallel to CD through B
<Line Tool>Construct line CE, intersecting circle A at F and circle B at G
<Parallel Tool>Construct the parallel to CE through D","['Circle Tool', 'Circle Tool', 'Line Tool', 'Parallel Tool', 'Parallel Tool', 'Line Tool', 'Parallel Tool']","[[], ['AC', 'B'], ['E', 'D', 'AB', 'C', 'BC', 'A'], ['CD'], [], [], ['F', 'G', 'CE'], []]",[]
Eta,https://euclidea.fandom.com/wiki/Sum_of_Areas_of_Squares,"Description: Construct a square whose area equals the sum of the areas of the two given squares and all three have the common angle.Let O be the common vertex of the squares, A the second vertex of one square on one line and B the second vertex of the other square on the other line.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Use the non-collapsing compass to construct the circle with radius AB and center O, intersecting one line at C and the other line at D
<Perpendicular Tool>Construct the perpendicular to OC from C
<Perpendicular Tool>Construct the perpendicular to OD from D","['Circle Tool', 'Perpendicular Tool', 'Perpendicular Tool']","[['O', 'A', 'B'], ['C', 'AB', 'D'], ['OC'], ['OD']]","['O', 'A', 'B']"
Eta,https://euclidea.fandom.com/wiki/Annulus,"Description: Construct a circle that is concentric with the given one and divides it into 2 parts of equal area.Let O be the center.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct an arbitrary line through O, intersecting the circle at A
<Perpendicular Tool>Construct the perpendicular bisector of OA, intersecting at B
<Circle Tool>Construct the circle with center B and radius OB, intersecting the bisector at C
<Circle Tool>Construct the circle with center O and radius OC","['Line Tool', 'Perpendicular Tool', 'Circle Tool', 'Circle Tool']","[['O'], ['A'], ['B', 'OA'], ['OB', 'C'], ['OC']]",['O']
Eta,https://euclidea.fandom.com/wiki/Angle_of_75%C2%B0,"Description: Construct an angle of 75° with the given side.Let A be the initial point of the given ray.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with an arbitrary center B not on the ray and radius AB, so that it intersects the ray at C and AC<AB
<Circle Tool>Construct the circle with center C and radius BC, intersecting circle B at D
<Angle Bisector Tool>Construct the angle bisector of  angle CAD","['Circle Tool', 'Circle Tool', 'Angle Bisector Tool']","[['A'], ['AC', 'AB', 'C', 'B'], ['BC', 'D'], ['CAD']]",['A']
Eta,https://euclidea.fandom.com/wiki/Line_Equidistant_from_Three_Points,"Description: Construct a line that is at equal distance from the given three points.Let A, B and C be the given points.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB
<Perpendicular Tool>Construct the perpendicular to AB through C, intersecting AB at D
<Perpendicular Tool>Construct the perpendicular bisector of CD","['Line Tool', 'Perpendicular Tool', 'Perpendicular Tool']","[['A', 'B', 'C'], ['AB'], ['D'], ['CD']]","['A', 'B', 'C']"
Eta,https://euclidea.fandom.com/wiki/Heron%27s_Problem,"Description: Construct a point C on the given line and segments AC and BC such that the sum of their length is minimal.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a circle with an arbitrary center D on the line and radius AD
<Line Tool>Construct a circle with an arbitrary center E on the line and radius AE, intersecting circle D at F
<Line Tool>Construct line BF, intersecting the line at C
<Line Tool>Construct line AC","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['C', 'AC', 'BC'], ['AD', 'D'], ['F', 'AE', 'E'], ['BF'], []]","['C', 'AC', 'BC']"
Eta,https://euclidea.fandom.com/wiki/Circumscribed_Circle,"Description: Construct the circumcircle of the triangle.Given the triangle ABC
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB
<Perpendicular Tool>Construct the perpendicular bisector of BC, intersecting the perpendicular bisector of AB at D
<Circle Tool>Construct the circle with center D and radius AD","['Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool']","[['ABC'], ['AB'], ['BC', 'D'], ['AD']]",['ABC']
Eta,https://euclidea.fandom.com/wiki/Inscribed_Circle,"Description: Construct the incircle of the triangle.Given the triangle ABC
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Angle Bisector Tool>Construct the angle bisector of  angle CAB
<Angle Bisector Tool>Construct the angle bisector of  angle ABC, intersecting the angle bisector of  angle CAB at D
<Perpendicular Tool>Construct the perpendicular to AB through D, intersecting AB at E
<Circle Tool>Construct the circle with center D and radius DE","['Angle Bisector Tool', 'Angle Bisector Tool', 'Perpendicular Tool', 'Circle Tool']","[['ABC'], ['CAB'], ['D'], ['AB', 'E'], ['DE']]",['ABC']
Eta,https://euclidea.fandom.com/wiki/Circle_Tangent_to_Three_Lines,"Description: Construct a circle that is tangent to the three given lines. Two of the lines are parallel.Given the intersections of the lines A and B.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Angle Bisector Tool>Construct the angle bisector of  angle CAB, with C an arbitrary point on one of the parallel lines.
<Angle Bisector Tool>Construct the angle bisector of  angle ABD, with D an arbitrary point on the other parallel line and on the same side as C from the third given line, intersecting the angle bisector of  angle CAB at E
<Perpendicular Tool>Construct the perpendicular to AB through E, intersecting AB at F
<Circle Tool>Construct the circle with center E and radius EF","['Angle Bisector Tool', 'Angle Bisector Tool', 'Perpendicular Tool', 'Circle Tool']","[['A', 'B'], ['CAB', 'C'], ['E', 'D', 'ABD'], ['F', 'AB'], ['EF']]","['A', 'B']"
Eta,https://euclidea.fandom.com/wiki/Segment_by_Midpoint,"Description: Construct a segment with the ends on the sides of the angle such that the given point is its midpoint.Let A be the vertex of the angle and B the midpoint.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Parallel Tool>Construct parallel of one ray through B, intersecting the other ray at C
<Circle Tool>Construct the circle with center C and radius AC, intersecting AC at D
<Line Tool>Construct line BD","['Parallel Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['C'], ['D', 'AC'], ['BD']]","['A', 'B']"
Eta,https://euclidea.fandom.com/wiki/Angle_Isosceles,"Description: Construct a line through the given point such that it cuts off equal segments on the sides of the angle.Let A be the vertex of the angle and B the point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Angle Bisector Tool>Construct the angle bisector A1 of the given angle
<Perpendicular Tool>Construct the perpendicular to A1 through B","['Angle Bisector Tool', 'Perpendicular Tool']","[['A', 'B'], [], []]","['A', 'B']"
Eta,https://euclidea.fandom.com/wiki/Excircle,"Description: Construct the excircle of the triangle formed by the three given lines.Let A, B and C be the intersections of the lines.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Angle Bisector Tool>Construct the angle bisector A1 of  angle DAB, with D an arbitrary point on the line that intersects AB at A
<Angle Bisector Tool>Construct the angle bisector A2 of  angle EBA, with E an arbitrary point on the line that intersects AB at B, intersecting A1 at F
<Perpendicular Tool>Construct the perpendicular to AB through F, intersecting AB at G
<Circle Tool>Construct the circle with center F and radius FG","['Angle Bisector Tool', 'Angle Bisector Tool', 'Perpendicular Tool', 'Circle Tool']","[['A', 'B', 'C'], ['DAB', 'AB', 'D'], ['F', 'EBA', 'E'], ['G'], ['FG']]","['A', 'B', 'C']"
Theta,https://euclidea.fandom.com/wiki/Perimeter_Bisector,"Description: Construct a straight line through a vertex of the triangle that divides its perimeter in half.Let A, B, and C be the vertices of the triangle
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a circle with center A and radius BC intersecting line AC at D
<Line Tool>Construct a circle with center C and radius AB intersecting line AC at E
<Perpendicular Tool>Construct the perpendicular bisector of DE intersecting line AC at F
<Line Tool>Construct the line BF","['Line Tool', 'Line Tool', 'Perpendicular Tool', 'Line Tool']","[['A', 'B', 'C'], ['AC', 'BC', 'D'], ['AB', 'E'], ['F', 'DE'], ['BF']]","['A', 'B', 'C']"
Theta,https://euclidea.fandom.com/wiki/Angle_54%C2%B0_Trisection,"Description: Construct two rays that divide the given angle of 54° into three equal parts.Let A be the vertex.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a circle with center B on either line and radius AB. Circle B intersects with line AB at point C, and the other line at point D.
<Circle Tool>Construct a circle with center D and radius AD. Circle D intersects with circle B at point E.
<Line Tool>Draw line AE, the first angle trisector.
<Angle Bisector Tool>Bisect angle DAE, the second angle trisector.","['Line Tool', 'Circle Tool', 'Line Tool', 'Angle Bisector Tool']","[['A'], ['D', 'AB', 'C', 'B'], ['AD', 'E'], ['AE'], ['DAE']]",['A']
Theta,https://euclidea.fandom.com/wiki/Interior_Angles,"Description: Construct a line through the point that crosses the two lines so that the interior angles are equal.Let A be the point of intersection between the two lines.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Point Tool> Let B be an arbitrary point on one of the lines, and C an arbitrary point on the other. Let O be the given point.
<Angle Bisector Tool>Bisect the angle BAC. The new line contains an arbitrary point D.
<Perpendicular Tool>Construct the line perpendicular to AD through O.","['Point Tool', 'Angle Bisector Tool', 'Perpendicular Tool']","[['A'], ['B', 'C', 'O'], ['BAC', 'D'], ['AD']]",['A']
Theta,https://euclidea.fandom.com/wiki/Regular_Octagon,"Description: Construct a regular octagon with the given side.Let point A and point B be the two given points.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular to line AB through point A. Let point C be an arbitrary point on this new line (C below AB to match orientation of explore mode).
<Line Tool>Construct the angle bisector of BAC. Let point D be an arbitrary point on this line.
<Perpendicular Tool>Construct the perpendicular to line AD through point B. Let point E be an arbitrary point on this line on the opposite side of point B from point C.
<Line Tool>Construct the angle bisector of ABE. This line intersects with line AD at point F and line AC at point G.
<Perpendicular Tool>Construct the perpendicular to line AB through point F.
<Perpendicular Tool>Construct the perpendicular to line AG through point G. This line intersects with line BE at point H.
<Perpendicular Tool>Construct the perpendicular bisector of point B and point H. This line intersects with line BG at point I.
<Perpendicular Tool>Construct the perpendicular of line AB through point I.
<Perpendicular Tool>Construct the perpendicular of line AD through point G (total 29E).","['Perpendicular Tool', 'Line Tool', 'Perpendicular Tool', 'Line Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool']","[['A', 'B'], ['AB', 'C'], ['BAC', 'D'], ['AD', 'E'], ['F', 'G', 'AC', 'ABE'], [], ['BE', 'H', 'AG'], ['I', 'BG'], [], []]","['A', 'B']"
Theta,https://euclidea.fandom.com/wiki/Triangle_Cleaver,"Description: Construct a line through the midpoint of a side of the triangle that bisects its perimeter.Let A, B, and C be the vertices of the triangle
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AC intersecting line AC at midpoint D
<Line Tool>Construct the line BE that bisects the angle ABC
<Line Tool>Draw the line parallel to BE that intersects D","['Perpendicular Tool', 'Line Tool', 'Line Tool']","[['A', 'B', 'C'], ['AC', 'D'], ['BE', 'ABC'], []]","['A', 'B', 'C']"
Theta,https://euclidea.fandom.com/wiki/Torricelli_Point,"Description: Construct a point from which each side subtends an angle of 120 or 60.Let A, B, and C be the vertices of the triangle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle A with center A and radius AB.
<Circle Tool>Construct circle B with center B and radius BA, intersecting circle A at D.
<Line Tool>Construct line CD, intersecting circle B at E.
<Perpendicular Tool>Construct line A as the perpendicular bisector of AE, intersecting line CD at F
<Point Tool>Select point F using the point tool.","['Circle Tool', 'Circle Tool', 'Line Tool', 'Perpendicular Tool', 'Point Tool']","[['A', 'B', 'C'], ['AB'], ['D', 'BA'], ['E', 'CD'], ['F', 'AE'], []]","['A', 'B', 'C']"
Theta,https://euclidea.fandom.com/wiki/Circle_Equidistant_from_Four_Points,"Description: Construct a circle equidistant from the given four points so that exactly two of them lie inside the circle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB
<Perpendicular Tool>Construct the perpendicular bisector of CD intersecting the previous line at E
<Line Tool>Construct the line AE
<Circle Tool>Construct a circle with center E and radius EC, intersecting AE at F
<Perpendicular Tool>Construct the perpendicular bisector of AF, intersecting AE at G
<Circle Tool>Construct a circle with center E and radius EG","['Perpendicular Tool', 'Perpendicular Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Tool', 'Circle Tool']","[[], ['AB'], ['E', 'CD'], ['AE'], ['F', 'EC'], ['G', 'AF'], ['EG']]",[]
Iota,https://euclidea.fandom.com/wiki/Minimum_Perimeter_-_1,"Description: Construct a triangle whose perimeter is the minimum possible whose vertices lie on the two sides of the angle and the third vertex is in the point A.Let O be the vertex of the angle and A the given point. Let B, C be abitary points on each ray, such that AB is bigger than AO and  AC is bigger than AO.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle O with center O and radius OA.
<Circle Tool>Construct circle B with center B and radius BA, intersecting circle O at F.
<Circle Tool>Construct circle B with center C and radius CA, intersecting circle O at G.
<Line Tool>Construct line FG, intersecting line OB at H, intersecting line OC at I.
<Line Tool>Construct line AH.
<Line Tool>Construct line AI.","['Circle Tool', 'Circle Tool', 'Circle Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['A', 'O', 'A', 'B', 'C', 'AB', 'AO', 'AC', 'AO'], ['OA'], ['F', 'BA'], ['G', 'CA'], ['OB', 'OC', 'FG', 'H', 'I'], ['AH'], ['AI']]","['A', 'O', 'A', 'B', 'C', 'AB', 'AO', 'AC', 'AO']"
Iota,https://euclidea.fandom.com/wiki/Third_Proportional,"Description: Construct a point D on the ray such that the segment AD is the third proportional to the given line segments.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle A with center A and radius AB.
<Circle Tool>Construct circle C with center C and radius CA, intersecting circle A at E.
<Circle Tool>Construct circle E with center E and radius EA, intersecting circle O at D.","['Circle Tool', 'Circle Tool', 'Circle Tool']","[['D', 'AD'], ['AB', 'A'], ['CA', 'E', 'C'], ['EA', 'O']]","['D', 'AD']"
Iota,https://euclidea.fandom.com/wiki/Harmonic_Mean_of_Trapezoid_Bases,"Description: Join the legs of the trapezoid with a segment that is parallel to the bases and whose length is equal to the harmonic mean of their length.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC.
<Line Tool>Construct line BD, intersecting line AC at E.
<Line Tool>Construct line parallel to AB through E.","['Line Tool', 'Line Tool', 'Line Tool']","[[], ['AC'], ['BD', 'E'], ['AB']]",[]
Iota,https://euclidea.fandom.com/wiki/Drop_a_Perpendicular*,"Description: Drop a perpendicular from the point to the line passing through the center of the circle using only a straightedge.Let O be the center of the circle, A be the given point, and B and C be the intersections of the circle and line.
Tool List: ['Point Tool', 'Line Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB, intersecting the circle at D.
<Line Tool>Construct line CD.
<Line Tool>Construct line AC, intersecting the circle at E.
<Line Tool>Construct line BE, intersecting CD at F.
<Line Tool>Construct line AF.","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['O', 'A', 'B', 'C'], ['AB', 'D'], ['CD'], ['E', 'AC'], ['BE', 'F'], ['AF']]","['O', 'A', 'B', 'C']"
Iota,https://euclidea.fandom.com/wiki/Midpoint*,"Description: Construct the midpoint of the segment using only a straightedge. To help you, a line parallel to the segment is given.Given the line AB.
Tool List: ['Point Tool', 'Line Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a line connecting A with an arbitrary point C on the line.
<Line Tool>Construct a line connecting B with another arbitrary point D on the line that intersects AC at E.
<Line Tool>Construct line AD.
<Line Tool>Construct line BC, intersecting AD at F.
<Line Tool>Construct line EF, intersecting AB at G.
<Point Tool>Point at G.","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Point Tool']","[['AB'], ['C', 'A'], ['D', 'E', 'AC', 'B'], ['AD'], ['F', 'BC'], ['EF', 'G'], []]",['AB']
Iota,https://euclidea.fandom.com/wiki/Trisection_by_Trapezoid_Diagonals,"Description: 3L 5E 2V
Tool List: ['Line Tool']
Solution: ","<Line Tool>Construct line AF, intersecting BD at G.
<Line Tool>Construct the line perpendicular to the bisector through G.","['Line Tool', 'Line Tool']","[[], ['AF', 'G', 'BD'], []]",[]
Iota,https://euclidea.fandom.com/wiki/Minimum_Perimeter_-_2,"Description: Inscribe a triangle into the given one whose perimeter is the minimum possible.Let A, B, and C be the vertices of the triangle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the line perpendicular to AB through C, intersecting AB at D.
<Perpendicular Tool>Construct the line perpendicular to AC through B, intersecting AC at E.
<Perpendicular Tool>Construct the line perpendicular to BC through A, intersecting BC at F.
<Line Tool>Construct line DE.
<Line Tool>Construct line DF.
<Line Tool>Construct line EF.","['Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Line Tool', 'Line Tool', 'Line Tool']","[['A', 'B', 'C'], ['AB', 'D'], ['E', 'AC'], ['F', 'BC'], ['DE'], ['DF'], ['EF']]","['A', 'B', 'C']"
Iota,https://euclidea.fandom.com/wiki/Harmonic_Mean_of_Segments,"Description: Construct a point on the given ray to get a segment with length equal to the harmonic mean of the two adjacent segments.Given line AB, let C be the intersection of AB and the other line.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle C with radius AC, intersecting line C at D and AB at E.
<Circle Tool>Construct circle D with radius DE.
<Line Tool>Construct line DE, intersecting circle D at F.
<Line Tool>Construct line BF, intersecting CD at G.
<Point Tool>Point at G.","['Circle Tool', 'Circle Tool', 'Line Tool', 'Line Tool', 'Point Tool']","[['AB', 'C', 'AB'], ['AC', 'E', 'D'], ['DE'], ['F'], ['G', 'BF', 'CD'], []]","['AB', 'C', 'AB']"
Iota,https://euclidea.fandom.com/wiki/Triangle_by_Angle_and_Centroid,"Description: Construct a segment connecting the sides of the angle to get a triangle whose centroid is in the point O.Let O be the vertex of the angle and A the given point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct the line parallel to one line through A, intersecting the other line at B.
<Line Tool>Construct the line parallel to BO through A, intersecting the other line at C.
<Circle Tool>Construct circle A with radius AB, intersecting AB at D.
<Circle Tool>Construct circle A with radius AC, intersecting AC at E.
<Line Tool>Construct line DE.","['Line Tool', 'Line Tool', 'Circle Tool', 'Circle Tool', 'Line Tool']","[['O', 'O', 'A'], ['B'], ['BO', 'C'], ['AB', 'D'], ['E', 'AC'], ['DE']]","['O', 'O', 'A']"
Iota,https://euclidea.fandom.com/wiki/Triangle_Mid-Segment,"Description: Construct a mid-segment of the given acute triangle.Let A, B, and C be the vertices of the triangle.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at D.
<Line Tool>Construct the line parallel to AC through D.","['Perpendicular Tool', 'Line Tool']","[['A', 'B', 'C'], ['AB', 'D'], ['AC']]","['A', 'B', 'C']"
Kappa,https://euclidea.fandom.com/wiki/Tangent_of_Circle,"Description: Construct a tangent to the circle from the external point.Let A be the center and B the given point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB.
<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at C.
<Circle Tool>Construct circle C with radius AC, intersecting circle A at D.
<Line Tool>Construct line BD.","['Line Tool', 'Perpendicular Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['AB'], ['C'], ['D', 'AC'], ['BD']]","['A', 'B']"
Kappa,https://euclidea.fandom.com/wiki/Outer_Tangent,"Description: Construct an outer tangent between two circles.Let A and B be the centers of the circles.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB, intersecting the two circles at C and D, with C and D between A and B.
<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at E.
<Perpendicular Tool>Construct the perpendicular bisector of CD, intersecting AB at F.
<Circle Tool>Construct circle E with radius AE, intersecting bisector F at G.
<Circle Tool>Construct circle G with radius CG, intersecting circle A at H and circle B at I.
<Line Tool>Construct line HI.","['Line Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['C', 'AB', 'D'], ['E'], ['F', 'CD'], ['G', 'AE'], ['H', 'CG', 'I'], ['HI']]","['A', 'B']"
Kappa,https://euclidea.fandom.com/wiki/Inner_Tangent,"Description: Construct an inner tangent between two circles.Let A and B be the centers of the circles.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB, intersecting circle A at C and circle B at D, with C on the other side of A from B and D between A and B.
<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at E.
<Perpendicular Tool>Construct the perpendicular bisector of CD, intersecting AB at F.
<Circle Tool>Construct circle E with radius AE, intersecting bisector F at G.
<Circle Tool>Construct circle G with radius CG, intersecting circle A at H and circle B at I.
<Line Tool>Construct line HI.","['Line Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['C', 'AB', 'D'], ['E'], ['F', 'CD'], ['G', 'AE'], ['H', 'CG', 'I'], ['HI']]","['A', 'B']"
Kappa,https://euclidea.fandom.com/wiki/Rotation_90%C2%B0,"Description: Rotate the line segment 90° counterclockwise about the given point.Given the segment AB and O the point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AO.
<Circle Tool>Construct circle O with radius AO.
<Perpendicular Tool>Construct the line perpendicular to AO through O, intersecting circle O at C.
<Circle Tool>Construct circle O with radius BO.
<Perpendicular Tool>Construct the line perpendicular to AB through C, intersecting the latest circle at D.
<Point Tool>Point at D.","['Line Tool', 'Circle Tool', 'Perpendicular Tool', 'Circle Tool', 'Perpendicular Tool', 'Point Tool']","[['AB', 'O'], ['AO'], [], ['C'], ['BO'], ['D'], []]","['AB', 'O']"
Kappa,https://euclidea.fandom.com/wiki/Rotation_60%C2%B0,"Description: Rotate the line segment 60° clockwise about the given point.Given the segment AB and O the point.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle A with radius AO.
<Circle Tool>Construct circle B with radius BO, intersecting circle A at C.
<Circle Tool>Construct circle O with radius AO, intersecting circle A at D, with D being clockwise around O from A.
<Line Tool>Construct line CD, intersecting circle B at E.
<Point Tool>Point at E.","['Circle Tool', 'Circle Tool', 'Circle Tool', 'Line Tool', 'Point Tool']","[['AB', 'O'], ['AO', 'A'], ['BO', 'B', 'C'], ['D'], ['E', 'CD'], []]","['AB', 'O']"
Kappa,https://euclidea.fandom.com/wiki/Segment_Trisection,"Description: Construct two points that divide the segment into 3 equal parts.Given the segment AB.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct circle A with radius AB.
<Perpendicular Tool>Construct the perpendicular bisector of AB, intersecting AB at C and circle A at D.
<Perpendicular Tool>Construct the perpendicular bisector of AD, intersecting CD at E.
<Perpendicular Tool>Construct the line perpendicular to BE through E, intersecting AB at F.
<Circle Tool>Construct circle C with radius CF, intersecting AB at G.
<Point Tool>Point at G.","['Circle Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Perpendicular Tool', 'Circle Tool', 'Point Tool']","[['AB'], ['A'], ['C', 'D'], ['AD', 'E', 'CD'], ['BE', 'F'], ['CF', 'G'], []]",['AB']
Kappa,https://euclidea.fandom.com/wiki/Segment_Trisection*,"Description: Construct points that divide the segment into three equal parts using only a straightedge. To help you, a line parallel to the segment is given.Given the segment AB.
Tool List: ['Point Tool', 'Line Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct a line connecting A with an arbitrary point C on the line.
<Line Tool>Construct a line connecting B with another arbitrary point D on the line that intersects AC at E.
<Line Tool>Construct line AD.
<Line Tool>Construct line BC, intersecting AD at F.
<Line Tool>Construct line EF, intersecting CD at G.
<Line Tool>Construct line AG, intersecting BD at H.
<Line Tool>Construct line FH, intersecting AB at I and CD at J.
<Line Tool>Construct line EJ, intersecting AB at K.
<Point Tool>Point at I.
<Point Tool>Point at K.","['Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Line Tool', 'Point Tool', 'Point Tool']","[['AB'], ['C', 'A'], ['D', 'E', 'AC', 'B'], ['AD'], ['F', 'BC'], ['EF', 'G', 'CD'], ['H', 'BD', 'AG'], ['FH', 'I', 'J'], ['EJ', 'K'], [], []]",['AB']
Kappa,https://euclidea.fandom.com/wiki/Chord_Trisection,"Description: Construct a chord of the larger circle through the given point that is divided into three equal segments by the smaller concentric circle.Given the center A and point on the larger circle B.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AB, intersecting the smaller circle at C and D and the larger circle at E.
<Circle Tool>Construct circle E with radius CD, intersecting the smaller circle at F.
<Line Tool>Construct line BF.","['Line Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['C', 'AB', 'D', 'E'], ['F', 'CD'], ['BF']]","['A', 'B']"
Kappa,https://euclidea.fandom.com/wiki/Three_Circles_-_1,"Description: Construct three circles with the given centers that are externally tangent each other.Given the points A, B, and C.
Tool List: ['Line Tool']
Solution: ",<Line Tool>Construct line AB.,['Line Tool'],"[['A', 'B', 'C'], ['AB']]","['A', 'B', 'C']"
Kappa,https://euclidea.fandom.com/wiki/Secant_Bisection,"Description: Construct a secant through the given point that is divided into two equal segments by the circle.Given the point A and centre B.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct the line AB intersecting the circle at C and D.
<Circle Tool>Construct the circle with centre B and radius AB, intersecting line AB at E.
<Circle Tool>Construct the circle with centre E and radius CD, intersecting circle B at F.
<Line Tool>Construct the line AF.","['Line Tool', 'Circle Tool', 'Circle Tool', 'Line Tool']","[['A', 'B'], ['C', 'AB', 'D'], ['E'], ['F', 'CD'], ['AF']]","['A', 'B']"
Kappa,https://euclidea.fandom.com/wiki/Three_Circles_-_2,"Description: Construct three circles tangent to each other at the given points.Given the points A, B, and C.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AB.
<Perpendicular Tool>Construct the perpendicular bisector of AC, intersecting the previous bisector at D.
<Line Tool>Construct the line AD.
<Perpendicular Tool>Construct the perpendicular to AD through A, intersecting the bisector of AB at E and the bisector of AC at F.
<Circle Tool>Construct the circle with centre E and radius AE.
<Circle Tool>Construct the circle with centre F and radius AF.
<Line Tool>Construct the line BE.
<Line Tool>Construct the line CF, intersecting BE at G.
<Circle Tool>Construct the circle with centre G and radius BG.","['Perpendicular Tool', 'Perpendicular Tool', 'Line Tool', 'Perpendicular Tool', 'Circle Tool', 'Circle Tool', 'Line Tool', 'Line Tool', 'Circle Tool']","[['A', 'B', 'C'], ['AB'], ['AC', 'D'], ['AD'], ['F', 'E'], ['AE'], ['AF'], ['BE'], ['CF', 'G'], ['BG']]","['A', 'B', 'C']"
Kappa,https://euclidea.fandom.com/wiki/Center_of_Rotation,"Description: Construct the center of rotation that transforms one of the given equal segments to the other one.Given one line AB and the other CD.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Perpendicular Tool>Construct the perpendicular bisector of AC.
<Perpendicular Tool>Construct the perpendicular bisector of BD, intersecting the other bisector at E.
<Point Tool>Point at E","['Perpendicular Tool', 'Perpendicular Tool', 'Point Tool']","[['AB', 'CD'], ['AC'], ['BD', 'E'], []]","['AB', 'CD']"
Lambda,https://euclidea.fandom.com/wiki/Fourth_Proportional,"Description: Construct a point D on the ray OC such that the segment OD is the fourth proportional to the given segments.Given points O, A, B, and C.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Line Tool>Construct line AC.
<Line Tool>Construct a line parallel to AC passing through B, intersecting ray OC at point D.
<Point Tool>Mark point D.","['Line Tool', 'Line Tool', 'Point Tool']","[['D', 'OC', 'OD', 'O', 'A', 'B', 'C'], ['AC'], [], []]","['D', 'OC', 'OD', 'O', 'A', 'B', 'C']"
Lambda,https://euclidea.fandom.com/wiki/Geometric_Mean_of_Segments,"Description: Construct a point on the given ray to get a segment with length equal to the geometric mean of two adjacent segments.Given line segment AB and ray O, where ray O extends from point O and |AO| = a and |BO| = b.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center O and radius |OA|, intersecting segment AB at C.
<Circle Tool>Construct a circle with center C and radius |CO|, intersecting circle O at D and E.
<Line Tool>Construct line DE.
<Circle Tool>Construct a circle with center B and radius |BO|, intersecting circle B at F.
<Circle Tool>Construct a circle with center O and radius |OF|, intersecting ray O at G.
<Point Tool>Mark the point G.","['Circle Tool', 'Circle Tool', 'Line Tool', 'Circle Tool', 'Circle Tool', 'Point Tool']","[['AB', 'O', 'O', 'O', 'AO', 'BO'], ['C', 'OA'], ['CO', 'E', 'D'], ['DE'], ['F', 'B'], ['OF', 'G'], []]","['AB', 'O', 'O', 'O', 'AO', 'BO']"
Lambda,https://euclidea.fandom.com/wiki/Golden_Section,"Description: Construct a point that divides the segment in the ""golden section"".Given line segment with points A and B.
Tool List: ['Point Tool', 'Line Tool', 'Circle Tool', 'Perpendicular Bisector Tool', 'Perpendicular Tool', 'Angle Bisector Tool', 'Parallel Tool', 'Compass Tool', 'Intersect Tool']
Solution: ","<Circle Tool>Construct a circle with center A and radius AB.
<Perpendicular Tool>Construct the perpendicular bisector of AB, marking intersection with AB as point C, and marking intersection with circle A as point D.
<Circle Tool>Construct a circle with center C and radius AB, marking intersection with circle A at E. E should be on the opposite side of AB from D, or else the solution will not work.
<Line Tool>Construct line DE, intersecting with AB at point F.
<Point Tool>Mark the point F.","['Circle Tool', 'Perpendicular Tool', 'Circle Tool', 'Line Tool', 'Point Tool']","[['A', 'B'], ['AB'], ['C', 'D'], ['E'], ['F', 'DE'], []]","['A', 'B']"