recursal/Fanatic-Fandom huggingface dataset:
Erect a Perpendicular

What we know is not much. What we do not know is immense. — Pierre-Simon Laplace

## Level Information

### Instruction:

Erect a perpendicular from the point on the line.

## 1L solution

Let O be the point, and A and B two arbitrary points on the line so that O is between them.

1. Construct the angle bisector of ∠AOB

## 3E solution

Let O be the point.

1. Construct a circle with an arbitrary center A not on the line and radius OA, intersecting the line at B
2. Construct line AB, intersecting circle A at C
3. Construct line OC

## Explanation

Solution 1L is simple as a line is an angle of 180°, so its angle bisector will divide it into two angles of 90° and will therefore be perpendicular to it. Just have the vertex of the 180° angle be point O.

Solution 3E makes use of Thales's theorem. You are constructing the inscribed triangle OBC with its side BC a diameter of the circle, which makes OBC a right triangle with ∠BOC=90°.
Equilateral Triangle

All three angles of an equilateral triangle are always 60° — Hint

## Level Information

### Instruction:

Construct an equilateral triangle with the given side. The construction must be exact. Guessing even very close points does not count as a solution.

## Tutorial

1. Construct a circle as displayed. Drag your finger from the left point directly to the right point until the circle sticks to it.
2. Construct the second circle of the same radius but with the center in the right point.
3. Select the Line tool and construct a straight line as shown. Drag your finger from the left point to the upper intersection of circles until the line sticks to it.
4. Construct the third side of the equilateral triangle.
5. You get a V-star if all the possible objects that satisfy the statement of a problem are constructed simultaneously. Construct the second equilateral triangle with a vertex that looks down.

## Solutions

Let A and B be the two end points of the given segment.

### 4L 4E Solution

1. Construct a circle with center A and radius AB
2. Construct a circle with center B and radius AB, intersecting circle A at points C and D
3. Construct line AC
4. Construct line BC

### 2V Solution

Starting with the 4L 4E solution,

1. Construct line AD
2. Construct line BD

## Explanation

A circle contains all points at an equal distance from the center. Since B and C both lie on the circle centered at A, AB=AC; likewise, since A and C both lie on the circle centered at B, AB=BC. As all three sides have the same length, the triangle ABC thus created is equilateral. The same is true for the second triangle ABD.
Shift Angle

Geometry is the real life. — Oscar Zariski

## Level Information

### Instruction:

Construct an angle from the given point that is equal to the given angle so that their sides are parallel.

## 2L solution

Let O be the given point.

1. Construct the parallel to one side of the angle through O
2. Construct the parallel to the other side of the angle through O

## 6E solution

Let O be the given point and A the vertex of the angle.

1. Construct line OA
2. Construct the circle with center A and radius OA, intersecting OA at B, one side of the angle at C and the other side of the angle at D
3. Construct the circle with center C and radius BC, intersecting circle A at E
4. Construct the circle with center D and radius BD, intersecting circle A at F
5. Construct line OE
6. Construct line OF

Alternate solution:

1. Construct the circle with center A and radius OA, intersecting one side of the angle at B and the other side at C
2. Construct the circle with center O and radius OA
3. Construct the circle with center B and radius AB, intersecting circle O at D
4. Construct the circle with center C and radius AC, intersecting circle O at E
5. Construct line OD
6. Construct line OE

## Explanation

Completing the parallelogram with A and O forming one diagonal produces the shifted angle. For the first 6E solution, two parallel lines are constructed by sharing the same reflection of point O across point A, then reflecting B across AC to E, as well as across AD to F. For the alternate solution, the two parallel lines are constructed by rhombi with side length OA by sharing circle A and O.
Parallelogram by Three Vertices

Logic is invincible, because in order to combat logic it is necessary to use logic. — Pierre Boutroux

## Level Information

### Instruction:

Construct a parallelogram whose three of four vertices are given.

## 4L solution

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)

1. Construct line AB
2. Construct line BC
3. Construct the parallel to AB through C
4. Construct the parallel to BC through A

## 8E solution

Let A, B and C be the given vertices.

1. Construct line AB
2. Construct line BC
3. Construct the circle with center A and radius AC, intersecting AB at D
4. Construct the circle with center C and radius AC, intersecting BC at E
5. Construct the circle with center D and radius AD, intersecting circle C at F
6. Construct the circle with center E and radius CE, intersecting circle A at G
7. Construct line AG
8. Construct line CF

Alternative construction:

1. Construct line AB
2. Construct line BC
3. Construct the circle with center B and radius AB, intersecting AB at D
4. Construct the circle with center B and radius BC, intersecting BC at E
5. Construct the circle with center A and radius AE, intersecting the larger circle B at F (technically, any point X along AB except B, with radius EX)
6. Construct the circle with center C and radius CD, intersecting the smaller circle B at G (technically, any point Y along BC except B, with radius DY)
7. Construct line AG
8. Construct line CF

## 3V solution

Starting from the 4L solution:

1. Construct line AC
2. Construct the parallel to AC through B (6L total)

Starting from either 8E solution, with point H defined as the intersection between lines AG and CF:

1. Construct line AC
2. Construct circle A with radius AH, intersecting line AG at I
3. Construct line BI (11E total)

## Explanation

The 4L solution minimizes moves by directly using two parallel lines, which is the defining characteristic of a parallelogram.

In the first 8E solution, E steps are minimized by sharing two circles when constructing parallel lines via two rhombi with edge-length AC (first in steps 3,4,5; second in 3,4,6). In the second 8E solution, E steps are minimized by reusing the first two lines of the parallelogram as the diameter of the circle used to form a right angle at the reflection of a point across a line (diameter CE used when reflecting E to F across AB; diameter AD used when reflecting D to G across BC).

The 3V solutions result from the fact that there are three choices of which two pairs of points to use for the first two lines. The minimum E solution determines the third parallel by symmetry to the first parallelogram (reflecting across the perpendicular to line AG), rather than by repeating the steps to construct another parallel from scratch.
Circle Center

Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. — Blaise Pascal

## Solutions

### 2L Solution

1. Construct the perpendicular bisector P1 of two arbitrary points on the circle.
2. Construct the perpendicular bisector of two other arbitrary points on the circle, intersecting P1 at point A.
3. Point at A.

### Alternative 2L Solution

1. Construct the perpendicular bisector of two arbitrary points on the circle, intersecting the circle at points A and B.
2. Construct the perpendicular bisector of AB, intersecting AB at C.
3. Point at C.

### 5E Solution

1. Construct a circle with an arbitrary center A on the circle and an arbitrary radius, intersecting the given circle at points B and C.
2. Construct a circle with center B and radius AB, intersecting circle A at points D and E.
3. Construct a circle with center C and radius AC, intersecting circle A at points F and G.
4. Construct line DE.
5. Construct line FG, intersecting DE at point H.
6. Point at H.

## Explanation

Given any two points on a circle, the perpendicular bisector of the segment made up of those two points passes through the center of the circle. Therefore, by constructing two perpendicular bisectors, you'll find the center.
The 5E solution uses the same reasoning, except you need to construct the perpendicular bisector without using the Perpendicular Bisector tool. As a reminder, in order to build a perpendicular bisector, you just need to construct two circles of equal radius with their centers on the circle and construct the line that passes through the intersections of those two circles. The trick to save 1 Elementary operation is to use one of the existing circles to construct your second perpendicular bisector.
Angle Bisector Tool

Note that the Angle Bisector tool costs 4 elementary (E) moves. — Euclidea Manual

## Level Information

### Instruction:

You have discovered the Angle Bisector tool. Click on any point on a side, on the vertex, and on any point on the other side.

## Solutions

Let A be the vertex of the angle between ray B and C.

### 1L 4E Solution

1. With the angle bisector tool active, select one of the two rays, then select the vertex, then select the other ray.

## Explanation

The angle bisector tool works with either three clicks (ray, vertex, ray); or with a click on a ray, then click-and-drag on the vertex and release on the final ray; or with a click-and-drag on the first ray with a release on the vertex, followed by a click on the final ray. Before the final action, the tool gives a visualization of where the bisector will appear. The angle bisector tool works even for a vertex of a 180° angle (that is, when both rays are collinear), although in such cases, it costs more elementary moves than using the Perpendicular tool. When selecting a ray, the tool prefers lines, but can also select points; a dashed line is drawn when there is not already a ray from a selected point to the vertex, although that decoration is not usable by the Point or Intersection tool.
Parallelogram by Three Midpoints

## 7L solution

1. Use the non-collapsing compass to construct the circle with radius AC and center B
2. 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
3. Construct line CD
4. Construct the parallel to CD through A
5. Construct the parallel to CD through B
6. Construct line CE, intersecting circle A at F and circle B at G
7. Construct the parallel to CE through D

## 10E solution

1. Construct line BC
2. Construct a circle with center C and radius BC, intersecting BC at D
3. Construct line AD
4. Construct a circle with center B and radius BC, intersecting BC at E
5. Construct a circle with center A and radius AC, intersecting circle B at F
6. Construct line EF
7. Construct line AF, intersecting circle A at G
8. Construct line CG, intersecting AD at H
9. Construct a circle with center C and radius CH, intersecting CG at I
10. Construct line BI

## 3V solution

From the 7L solution:

1. Construct line BD
2. Construct line BF
3. Construct the parallel to BF through A
4. Construct the parallel to BF through C
5. Construct the parallel to BD through F
6. Construct line AD
7. Construct line AG
8. Construct the parallel to AG through B
9. Construct the parallel to AG through C
10. Construct the parallel to AD through G
Point Reflection

## 4L solution

Given the segment AB and O the point.

1. Construct line AO
2. Construct the circle with center O and radius AO, intersecting AO at C
3. Construct the parallel to AB through C
4. Construct line BO, intersecting the parallel to AB at D
5. Point at D

## 5E solution

Given the segment AB and O the point.

1. Construct the circle with center O and radius AO
2. Construct the circle with center O and radius BO
3. Construct line AO, intersecting the first circle at C
4. Construct line BO, intersecting the second circle at D
5. Construct line CD
Rotation 90°

Instruction: Rotate the line segment 90° counterclockwise about the given point.

Goal: 5L 8E

Available Tools:

* Move
* Point
* Line
* Circle
* Perpendicular Bisector
* Perpendicular
* Angle Bisector
* Parallel
* Non-collapsing Compass
* Intersect

Pack: Kappa

Previous Level: 10.3

Next Level: 10.5

## 5L Solution

Given the segment AB and O the point.

1. Construct line AO.
2. Construct circle O with radius AO.
3. Construct the line perpendicular to AO through O, intersecting circle O at C.
4. Construct circle O with radius BO.
5. Construct the line perpendicular to AB through C, intersecting the latest circle at D.
6. Point at D.

## 8E Solution

Given the segment AB and O the point.

1. Construct line AB.
2. Construct circle O with radius AO, intersecting AB at C.
3. Construct line AO, intersecting circle O at D.
4. Construct circle C with radius CD, intersecting AB at E, with E on the other side of the solution than A.
5. Construct the perpendicular bisector of AE, intersecting circle O at one point in the solution.
6. Construct circle O with radius BO, intersecting the bisector at the other point in the solution.
Torricelli Point

## 4L solution

Let A, B, and C be the vertices of the triangle.

1. Construct circle A with center A and radius AB.
2. Construct circle B with center B and radius BA, intersecting circle A at D.
3. Construct line CD, intersecting circle B at E.
4. Construct line A as the perpendicular bisector of AE, intersecting line CD at F
5. Select point F using the point tool.

## 5E solution

Let A, B, and C be the vertices of the triangle.

1. Construct circle A with center A and radius AB.
2. Construct circle B with center B and radius BA, intersecting circle A at D.
3. Construct line CD, intersecting circle B at E.
4. Construct circle E with center E and radius EB, intersecting circle B at F.
5. Construct line AF, intersecting line CD at G.
6. Select point G with the point tool.

## 2V solution

Repeat steps 3-5 of the 4L solution, letting point D be the other intersection between circles A and B.

## Explanation

I could not follow the steps above. I looked it up, and found this page (https://demonstrations.wolfram.com/TheTorricelliPoint/). Apparently, if you construct equilateral triangles based on each edge, each line going from an outer vertex to the opposite vertex of the original triangle will go through this point.
CollegeHumor Chaos

***Join the CollegeHumor cast LIVE for original games, silly challenges, audience Q and A, rejected sketches, behind-the-scenes trivia, and who knows what else by the time you finish reading this?***

**CollegeHumor Chaos** is

## Episodes

### Season 1

* Episode 1: We Put On a Lot of Clothes
* Episode 2: Psychic Game Show
* Episode 3: Truth Tower
* Episode 4: Blow My Mind
Abbie Emmons

**BASIC INFORMATION**

Abbie is a self-published author who is also a YouTuber, singer/songwriter, blogger, traveller and filmmaker. She lives in Vermont, USA.

**AGE RANGE**

Young Adult

**GENRE**

Contemporary

**BOOKS**

- 100 days of sunlight (2019)

- Tessa and Weston: The Best Christmas Ever (2021)

**WEBSITES AND MORE INFORMATION**

https://abbieemmonsauthor.com

https://www.amazon.co.uk/Abbie-Emmons/e/B07RS9LBPF?ref=sr\_ntt\_srch\_lnk\_1&qid=1592819746&sr=1-1

https://www.waterstones.com/book/100-days-of-sunlight/abbie-emmons/9781733973311
Tangent to Circle at Point

If a man's wit be wandering, let him study the mathematics. — Francis Bacon

## 2L solution

Let O be the center of the circle and A the point on it.

1. Construct line OA
2. Construct the perpendicular to the line OA from point A

## 3E solution

Let O be the center of the circle and A the point on it.

1. Construct a circle with an arbitrary center B on the circle and radius AB, intersecting circle O at C
2. Construct the circle with center A and radius AC, intersecting circle B at D
3. Construct line AD

## Explanation

Solution 2L is easy as a tangent is simply a line perpendicular to a radius at a point on the circle.

Solution 3E is an application of Thales' Theorem. Let DA intersect the bigger circle again at E.
∠ECD is a right angle by Thales' theorem. Let EC intersect the smaller circle again at F.
Since CDAF is cyclic and ∠FCD = 90°, ∠FAD = 90° and it passes through the center. So, DE is a tangent.
Wiki rules

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Circle Tangent to Line

Intellectual labor in math lessons is the touchstone of thinking. — Vasili Sukhomlinsky

## Level Information

### Instruction:

Construct a circle with the given center that is tangent to the given line.

## 2L 4E solution

Let O be the center.

1. Construct the perpendicular to the line from point O, intersecting the line at A
2. Construct the circle with center O and radius OA

## Explanation

A tangent is a line perpendicular to a radius at the point of tangency on the circle.
Karen M. McManus

**BASIC INFORMATION**

Karen is the New York Times best-selling author of the young adult thriller and mystery story *One of Us is lying.* Her fourth book, titled *The cousins* is due to come out in December 2020.

**AGE**

Young Adult

**GENRE**

Mystery, thriller

**BOOKS**

- One of us is lying (2017)

- Two can keep a secret (2019)

- One of us is next (2020)

- The cousins (2020)

**WEBSITES AND MORE INFO**

https://www.waterstones.com/author/karen-mcmanus/228286
Angle of 75°

## 3L solution

Let A be the initial point of the given ray.

1. 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
2. Construct the circle with center C and radius BC, intersecting circle B at D
3. Construct the angle bisector of ∠CAD

## 5E solution

Let A be the initial point of the given ray.

1. Construct a circle with center A and an arbitrary radius, intersecting the ray at B
2. Construct the circle with center B and radius AB, intersecting circle A at C and D
3. Construct the circle with center C and radius AC
4. Construct line CD, intersecting circle C at E
5. Construct line AE

## 2V solution

From the 5E solution:

1. Construct line AC, intersecting circle A at G
2. Construct line BG, intersecting circle B at H
3. Construct line AH
Given Angle Bisector

## Level Information

### Instruction:

Construct two straight lines through the two given points respectively so that the given line is a bisector of the angle that they make.

## 4L 4E solution

Let A and B be the given points.

1. Construct a circle with an arbitrary center C on the line and radius AC
2. Construct a circle with another arbitrary center D on the line and radius AD, intersecting circle C at E
3. Construct line BE, intersecting the line at F
4. Construct line AF

## Explanation

Label the points A and B. Reflect A across the line to make A'. The line A'B is a reflection of AB.
Line Equidistant from Two Points - 2

Mathematics is the science of those things that are clear in and of themselves. — Carl Jacobi

## Level Information

### Instruction:

Construct a line through the point C that goes between the points A and B and that is at equal distance from them.

## 3L 5E solution

Given points A, B, and C, with C labeled and the choice of A and B arbitrary:

1. Construct the perpendicular bisector of AB
2. Construct line AB, intersecting its perpendicular bisector at D
3. Construct line CD

## Explanation

Two points are equidistant from any line drawn through the midpoint of those two points. Once the midpoint is determined, finish the construction by connecting it to point C.
Glendale Galleria/Tenant List

## Current

### Anchors

* Bloomingdale's
* Dick's Sporting Goods
* JCPenney
* Macy's
* Target

### Inline stores

* 77 Kids
* Abercrombie & Fitch/Abercrombie Kids
* Adidas
* Aerie
* Aeropostale/P.S. from Aeropostale
* ALDO
* American Eagle Outfitters
* Apple Store
* Banana Republic
* Bath & Body Works
* Build-A-Bear Workshop
* Claire's
* Clarks
* Coach
* The Children's Place
* Disney Store
* Express/Express Men
* Foot Locker
* Gap
* Godiva
* Gucci
* Guess
* Gymboree
* Hollister
* Hot Topic
* Justice & Brothers
* The Limited
* Limited Too
* L'Occitane
* Lucky Brand Jeans
* Michael Kors
* Oakley
* Skechers
* Spencer's
* Sunglass Hut
* Superdry
* Ted Baker London
* Tillys
* Uniqlo
* Victoria's Secret
* The Walking Company
* White House Black Market
* Zales Jewelers
* Zara

### Dining

* Friendly's
* Haagen-Dazs
* McDonald's

## Former

### Anchors

* Robinsons-May
* Nordstrom
Circle in Square

You can't fit a round peg in a square hole. — American proverb

## 1.4 Circle in Square

Given the square ABCD.

2. Construct the perpendicular bisector of AB, intersecting AB at point E and AC at point F.
3. Construct a circle with center E and radius EF.

## Explanation

In a square, the center of the inscribed circle is the intersection of its diagonal and the intersection of the perpendicular bisector of its sides. Constructing one diagonal and one perpendicular bisector is enough to find the center of the circle. One perpendiculor bisector is needed to determine the radius as the inscribed circle passes through all the midpoints of the square's sides.
Fourth Proportional

Mathematics is concerned only with the enumeration and comparison of relations. — Carl Friedrich Gauss

## Level Information

### Instruction:

Construct a point D on the ray OC such that the segment OD is the fourth proportional to the given segments.

## 2L solution

Given points O, A, B, and C.

1. Construct line AC.
2. Construct a line parallel to AC passing through B, intersecting ray OC at point D.
3. Mark point D.

## 3E solution

Given points O, A, B, and C.

1. Construct circle A with radius AB, intersecting OA at D.
2. Construct circle C with radius CD, intersecting circle A at D and E.
3. Construct line BE, intersecting ray O at F.
4. Mark point F.

## Explanation

The proportionality of line segments along lines originating from a single point passing through a pair of parallel lines is constant.
Midpoints of Trapezoid Bases

If there is a God, he is a great mathematician. — Paul Dirac

## 3L solution

Given the trapezoid ABCD.

1. Construct the perpendicular bisector of the base AB, intersecting AB at E
2. Construct the perpendicular bisector of the base CD, intersecting CD at F
3. Construct line EF

## 5E solution

Given the trapezoid ABCD.

1. Construct line AC, a diagonal of the trapezoid
2. Construct line BD, the other diagonal of the trapezoid, intersecting AC at E
3. Construct line AD
4. Construct line BC, intersecting AD at F
5. Construct line EF

## Explanation

Since the perpendicular bisector of a segment intersects its segment at its midpoint, you can construct the midpoints of the bases and join them, resolving the puzzle.

Solution 5E makes use of a property in a trapezoid where the intersection of the legs (the non-parallel sides), the midpoints of the two bases and the intersection of the diagonals are all collinear. This property can be demonstrated by the intercept theorem.

Considering the two (parallel) bases and the diagonals, the intercept theorem gives you EA/EC=EB/ED=AB/CD.

Let G be the midpoint of AB and H the intersection of EG and the base CD. The intercept theorem gives you EA/EC=EG/EH=AG/CH

Since 2AG=AB, 2(CH\*EA/EC)=CD\*EA/EC and therefore 2CH=CD, meaning H is also the midpoint of CD, making G, E and H collinear.

A similar demonstration considering the triangle CDF and the parallel lines AB and CD also shows that the midpoints of the two parallel segments and the vertex F of the triangle are also collinear.
Heron's Problem

## Level Information

### Instruction:

Construct a point C on the given line and segments AC and BC such that the sum of their length is minimal.

## 4L 4E solution

1. Construct a circle with an arbitrary center D on the line and radius AD
2. Construct a circle with an arbitrary center E on the line and radius AE, intersecting circle D at F
3. Construct line BF, intersecting the line at C
4. Construct line AC

## Explanation

Consider the given line as a mirror. Once A is reflected across the line to F, the shortest distance is then a straight line which hits the mirror at C.
Circle Tangent to Three Lines

## Level Information

### Instruction:

Construct a circle that is tangent to the three given lines. Two of the lines are parallel.

## 4L solution

Given the intersections of the lines A and B.

1. Construct the angle bisector of ∠CAB, with C an arbitrary point on one of the parallel lines.
2. Construct the angle bisector of ∠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 ∠CAB at E
3. Construct the perpendicular to AB through E, intersecting AB at F
4. Construct the circle with center E and radius EF

## 6E solution

Given the intersections of the lines A and B.

1. Construct the circle with center A and radius AB, intersecting the parallel line through A at C and D
2. Construct line BC
3. Construct the circle with center C and radius BC
4. Construct the circle with center D and radius CD, intersecting circle C at E and F
5. Construct line EF, intersecting BC at G and AC and H
6. Construct the circle with center G and radius GH

## 2V solution

From the 4L solution:

1. Construct the angle bisector of ∠GAB, with G an arbitrary point on AC and on the other side of C from A.
2. Construct the angle bisector of ∠ABH, with H an arbitrary point on BD and on the other side of D from B, intersecting the angle bisector of ∠GAB at I
3. Construct the perpendicular to AB through I, intersecting AB at J
4. Construct the circle with center I and radius IJ
Angle 54° Trisection

## 4L solution

Let A be the vertex.

1. 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.
2. Construct a circle with center D and radius AD. Circle D intersects with circle B at point E.
3. Draw line AE, the first angle trisector.
4. Bisect angle DAE, the second angle trisector.

Alternative 4L solution:

1. Construct a circle with center B on either line and radius AB, intersecting the other line at point C.
2. Constuct the parallel to AB through C, intersecting circle B at point D.
3. Constuct line AD, the first angle trisector.
4. Construct the angle bisector of ∠DAB, the second angle trisector.

## 5E solution

Let A be the point of intersection between the two lines

1. 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.
2. Construct a circle with center D and radius AD. Circle D intersects with circle B at point E.
3. Draw line AE. This is the first angle trisector.
4. Draw circle with center E and radius EC. Circle E intersects with circle B at point F.
5. Draw line AF. This is the second angle trisector.

## Explanation

As mentioned in the FAQ, trisection of an angle is not possible in general case, so you need to use that 54/3 = 18. You can also note that 90-54 = 36 = 2\*18 and so on.
Chord Parallel to Segment

## Level Information

### Instruction:

Construct a chord of the given circle that is parallel to the given segment and congruent to it.

## Location Information

### Previous Level:

8.9 Egyptian Triangle by Side of Length 4

## 4L solution

Let A and B be the given points on the line segment. Let circle C be the given circle with center point C.

1. Construct line C parallel to line AB through point C.
2. Construct circle D with center C and radius equal to the length of AB using the non-collapsing compass tool. It intersects with line C at two points, call one of them E.
3. Construct the perpendicular bisector of CE, intersecting circle C at point F.
4. Erect a perpendicular to the perpendicular bisector of CE at point F.

## 7E solution

Let A and B be the given points on the line segment. Let circle C be the given circle with center point C.

1. Draw the line AB.
2. Draw the circle D, with center point C and radius CB, intersecting line AB at B and E. (If B and E are very close together, use the Move tool to shift them slightly.)
3. Draw the perpendicular bisector of AE, intersecting circle C at F and G.
4. Draw the line FC, intersecting circle C at F and H.
5. Draw the line GH.

## 2V solution

Start from the 4L solution and let point G be the other intersection point between circle C and the perpendicular bisector of CE.

1. Erect a perpendicular to the perpendicular bisector of CD at point G.
Perpendicular Bisector

Education is what remains after one has forgotten everything he learned in school. — Albert Einstein

## Level Information

### Instruction:

Construct the perpendicular bisector of the segment.

## Location Information

### 3L 3E Solution

1. Construct a circle with center A and radius AB.
2. Construct a circle with center B and radius AB, intersecting circle A at C and D.
3. Construct line CD.

## Explanation

Any point of a perpendicular bisector is equidistant to both A and B. The two circles ensure that AC=BC and AD=BD and so both C and D are on the perpendicular bisector of segment AB.
Intersect Tool

Every intersection has a story. — Katherine Dunn

## Level Information

### Instruction:

If it is hard to hit the required intersection since lines are too close to each other, use the Intersect tool. Select one object (a circle or a line), then the second one. To practice, construct the intersection of the given circles.

## Solutions

### 0L 0E Solution

1. Use the intersection tool to select both circles, which then highlights both solution points.

## Explanation

The intersection tool can be an easier way to highlight particular points than attempting to zoom in and manually select which point is desired.
The Rank Room

***Host and philosopher Katie Marovitch recruits comedians to answer life’s important questions, like “What’s the sexiest way to die?”***

**The Rank Room** is

## Episodes

* Episode 1: What Is the Best Invention That Has Yet to Be Invented?
* Episode 2: What Is the Worst Marriage Proposal?
* Episode 3: What is the Coolest Accidental Death?
* Episode 4: What is the Best Thing to Find in a Treasure Chest?
* Episode 5: What Is the Worst Spell a Witch Could Curse You With?
* Episode 6: What Is the Worst Tattoo to Get?
* Episode 7: What Is the Coolest Way to Quit Your Job?
* Episode 8: What Is the Coolest Way to Murder Someone?
* Episode 9: What Is the Worst Way to Find Out You Were Cheated On?
* Episode 10: What Is the Best Funeral You Can Imagine?
* Episode 11: What Is the Coolest Way to Break Out of Jail?
* Episode 12: What Is the Best Way to Spend Lotto Money?
* Episode 13: What Is the Best Way to Spend Your Last Day on Earth?
* Episode 14: What Is The Worst Reason For Getting Famous?
* Episode 15: What Is the Best Way to Improve the Body?
* Episode 16: What Is the Coolest Family Secret?
* Episode 17: The Rank Room: Behind the Scenes
Square Root of 2

The square root of 2 is also known as Pythagoras' constant. — Wikipedia

## Level Information

### Instruction:

Let |AB|=1. Construct a point C on the ray AB such that the length of AC is equal to √2.

## 3L 5E solution

1. Construct the perpendicular to AB from B
2. Construct the circle with center B and radius AB, intersecting the perpendicular at D
3. Construct the circle with center A and radius AD, intersecting the ray at C
4. Point at C

## Explanation

∠

{\displaystyle \angle}

ABD = 90o (step 1), and AB = 1 = BD (step 2).

∴

{\displaystyle \therefore}

by using the 45-45-90 special triangle, we know that AD =

2

{\displaystyle \sqrt{2}}

.

AD and AC are both radii of the circle centered at A (step 3),

∴

{\displaystyle \therefore}

AC =

2

{\displaystyle \sqrt{2}}

as well.
Translate Segment

## Level Information

### Instruction:

Construct a segment from the given point parallel and equal to the given segment.

## 2L 2V solution

Given the segment AB and the point C.

1. Use the non-collapsing compass to construct the circle with radius AB and center C
2. Construct the parallel to AB through C, intersecting the circle at two points
3. Use the intersect tool on the circle and parallel line to select both points at once

## 6E solution

Given the segment AB and the point C.

1. Construct the circle with center A and radius AB
2. Construct the circle with center B and radius AB, intersecting circle A at D and E
3. Construct the circle with center D and radius CD
4. Construct the circle with center E and radius CE, intersecting circle D at F
5. Construct line CF
6. Construct the circle with center A and radius AF, intersecting line CF at G
7. Point at G

## 2V solution

From the 6E solution:

1. Construct the circle with center C and radius CG, intersecting CG at H
2. Point at H
Wiki rules

1. **Keep it civil**: Do not make personal attacks on other people. If you need to criticize another user’s argument, do so without attacking them as a person. Do not use bigoted language, including slurs which degrade another person or group of people based on gender, race, sexual orientation, nationality, religion, etc.
2. **Be a productive member of the wiki**: Contribute to the wiki in line with the established processes and conventions. Disrupting the wiki with “edit warring” does not help.
3. **Do not engage in excessive self-promotion**: The wiki is a collaborative community resource for the topic at hand. It is NOT a free place to advertise your related website, YouTube channel, blog, social media account, etc. Have a question about whether your link would be welcome? Ask an admin (Chemicalpenguin).
4. **Do not harass other users**: If somebody asks you to stop posting certain content on their wall, respect their wishes. It is their wall.
5. **Do follow community guidelines for formatting**: When a community has established formatting, it’s important to adhere to this.
6. **Keep it neutral**: When writing pages about authors, don't write about your favourite books (or least favourite!). While this site was created to discuss all things book-related, the pages on authors are merely there for people to use to learn. If you have an opinion or want to ask a question, leave it in the comments below the post!
7. **Don't relate posts to you**: If you know an author, please don't state this in the wiki - that's what the comments are for! As mentioned above, the only information you should be putting into the pages are facts about the author themselves.

If you have any questions, make sure to ask me (Chemicalpenguin).
Three Circles - 2

## 9L Solution

Given the points A, B, and C.

1. Construct the perpendicular bisector of AB.
2. Construct the perpendicular bisector of AC, intersecting the previous bisector at D.
3. Construct the line AD.
4. Construct the perpendicular to AD through A, intersecting the bisector of AB at E and the bisector of AC at F.
5. Construct the circle with centre E and radius AE.
6. Construct the circle with centre F and radius AF.
7. Construct the line BE.
8. Construct the line CF, intersecting BE at G.
9. Construct the circle with centre G and radius BG.

## 13E Solution

Given the points A, B, and C.

1. Construct the line AB
2. Construct the circle with centre A and radius AC, intersecting AB at D (with D closer to B than A).
3. Construct the circle with centre B and radius BC, intersecting AB at E (with E closer to A than B) and circle A at F.
4. Construct line DF, intersecting circle B at G.
5. Construct line EF, intersecting circle A at H.
6. Construct line BG.
7. Construct line AH, intersecting BG at I.
8. Construct circle with centre I and radius AI.
9. Construct line BF, intersecting circle A at J.
10. Construct circle with centre C and radius CJ, intersecting circle A at K.
11. Construct line CK, intersecting AH at L and BG at M.
12. Construct circle with centre L and radius AL.
13. Construct circle with centre M and radius BM.
Angle of 60°

Let no one ignorant of geometry enter here. — Plato

## Solutions

Let A be the initial point of the given ray.

### 3L 3E Solution

1. Construct a circle with center A and an arbitrary radius, intersecting the ray at point B.
2. Construct a circle with center B and radius AB, intersecting circle A at points C and D.
3. Construct line AC.

## Explanation

All internal angles of an equilateral triangle are 60°, so you just need to construct the two sides of an equilateral triangle adjacent to vertex A.

The two circles ensure that AB=AC=BC and AB=AD=BD and so ABC and ABD are equilateral and angles CAB and DAB are 60°.
Robin stevens

## Basic Information

Robin Stevens is a children's author, most famous for writing the Murder Most Unladylike series, following two girls, Hazel and Daisy, as they solve crimes together. She was born in California, but moved to England when she was three, growing up in Oxford. [1]

## Books

*Murder Most Unladylike Series:*

* Murder Most Unladylike (2014)
* Arsenic For Tea (2014)
* First Class Murder (2015)
* Jolly Foul Play (2016)
* Mistletoe and Murder (2016)
* Cream Buns and Crime (2017) *Short Story Collection*
* A Spoonful of Murder (2018)
* Death in the Spotlight (2018)
* Top Marks for Murder (2019)
* Death Sets Sail (2020)
* Once Upon a Crime (2021) *Short Story Collection*

Other Books:

* The Guggenheim Mystery (2017) (sequel to Siobhan Dowd's *The London Eye Mystery)*

* The Ministry of Unladylike Activity (2022)

## Awards

Table
 2015
 2016
 2017

## Resources

[1] https://robin-stevens.co.uk/

[2] https://en.wikipedia.org/wiki/Robin\_Stevens\_%28author%29
Bethany Atazadeh

**BASIC INFORMATION**

Bethany Atazadeh is a Minnesota-based author of young adult science-fiction and fantasy novels, children's books, and non-fiction, as well as a YouTuber.

**AGE**

Her books are mainly for children and young adults.

**BOOKS**

- The Stolen Kingdom (2019)

- The Jinny Key (2019)

- The Cursed Hunter (2020)

- The Enchanted Crown (not out yet)

**The Number Series:**

- Evalene's Number (2018)

- Pearl's Number (2018)

*Children's:*

- The Confident Corgi (2018)

- How your book sells itself (2019) in collaboratiion with Mandi Lynn

- Grow Your Author Platform (2019)

- Book sales that multiply (2019)

**WEBSITES AND MORE INFORMATION**

https://www.bethanyatazadeh.com/
Helena Duggan

##### Historia

Soy de Kilkenny, una ciudad medieval embrujada en el sur de Irlanda, que fue la inspiración para la ciudad en A Place Called Perfect. Escribo historias llenas de aventuras porque me aburro con mucha facilidad. También soy diseñador gráfico para el negocio de diseño de iluminación de mi hermano y hermanas Willie Duggan en Irlanda.

Un lugar llamado perfecto fue mi primer libro. Se publicó en  agosto  de 2017 y fue un Libro del Mes de Waterstones. Ganó numerosos premios, incluido  el Libro infantil  del año de Crimefest y fue nominado para los Premios del libro irlandés y el  Premio infantil Waterstones  .

La serie es una serie superventas que ha vendido más de 230.000 copias en el Reino Unido y  sigue contando y se publica en 13 idiomas hasta la fecha.  Perfect  también ha sido seleccionado  para  una serie de televisión animada por Jellyfish Productions.

The Light Thieves es mi nueva serie de aventuras ecológicas y tecnológicas que acaba de lanzarse con excelentes críticas. Espero escuchar cómo es  recibido por el mundo en general.

Cuando no estoy escribiendo o diseñando, estoy con Robbie, mi esposo (a quien le encanta escalar montañas), mis hijas Jo y Bobbie y nuestro loco perro de rescate Tinky.

**WEBSITES**

https://www.helenadugganwrites.com/
Midpoint*

*This page is for the level in Kappa. See Midpoint\* for the level in Omicron.*

## Level Information

### Instruction:

Construct the midpoint of the segment using only a straightedge. To help you, a line parallel to the segment is given.

## 5L 5E Solution

Given the line AB.

1. Construct a line connecting A with an arbitrary point C on the line.
2. Construct a line connecting B with another arbitrary point D on the line that intersects AC at E.
3. Construct line AD.
4. Construct line BC, intersecting AD at F.
5. Construct line EF, intersecting AB at G.
6. Point at G.
Kat Ellis

## ABOUT

Ellis studied English at Manchester Metropolitan University. When not writing, she explores ancient ruins and cemeteries around North Wales, or watching scary movies with her husband. She has written four novels and was part of a collection of three novellas.

## BOOKS

* Blackfin Sky (2014)
* Breaker (2016)
* Purge (2016)
* Harrow Lake (2020)

NOVELLAS

* Three Strikes (2018)

## AGE

Kat Ellis writes books for young adults/teenagers.

## WEBSITES AND MORE INFORMATION

https://katelliswrites.blogspot.com/

https://www.waterstones.com/author/kat-ellis/1182979
Aleksander Petit

## Basic Information

Publishing has always been Aleksander Petit's dream. Having always struggled with mental health issues, a primary aim as an author for him is to bring postive representation of mental health issues to those who need it.

## Genres

Aleks writes for young adults. His first trilogy, the Trick Questions series, falls into the sub-genre of spy-thriller.

## Books

THE TRICK QUESTIONS SERIES

* *Silver-Lined Secrets* (2022)

## Websites and more information

Aleksander Petit's Twitter: https://twitter.com/alekpetitwrites
Cassandra Clare

BASIC INFO

Judith Lewis, better known by her pen name Cassandra Clare, is an American author of young adult fiction.

BIRTHDAY

July 27, 1973, Tehran

AGE

Cassandra Clare writes for young adults

GENRE

Fantasy

MOST KNOWN FOR

One of her most renowned series is titled *the Shadowhunter Chronicles.*

WEBSITES AND MORE INFO

https://www.cassandraclare.com/
Circle Tool

The Circle Tool is introduced during the tutorial level Equilateral Triangle. When active, the tool creates circles by a compass; either by two consecutive clicks on the center and then a point on the circumference, or by clicking on the center and dragging to the desired radius. When selecting the center or circumference point, the tool tries to favor nearby intersections between lines (whether straight lines or the circumference of a circle) or points previously marked with the Point Tool; however, it is also possible for either defining point of a circle to be placed on a free point.

The Circle Tool costs one elementary move if it creates a new circle. There is no cost for trying to re-create a circle that has already been constructed. There are a few levels where a circle with a center on a free point is necessary for minimizing the E score, and many others where a circle with an arbitrary radius starts out the construction process.
CH Live

***Get a front row seat for live performances from classic CollegeHumor sketch and stand up shows, featuring some of today's best comedians before they were big.***

**CH Live** is

## Performers

* Pete Holmes
* Jacqueline Novak
* Christian Finnegan
* Jeff Rubin
* Josh Ruben

## Episodes

### Season 1

* Episode 1: Pete Holmes
* Episode 2: Jacqueline Novak
* Episode 3: Christian Finnegan
* Episode 4: Jeff Rubin
* Episode 5: Josh Ruben
Holly Jackson

## **BASIC INFORMATION**

Holly grew up in Buckinghamshire and started writing stories from a young age, completing her first attempt at a book aged fifteen. She now lives in London, England. Her debut novel is called *A Good Girl's Guide to Murder* and was published by Electric Monkey, an imprint of Egmont in the U.K.

## **BOOKS**

A GOOD GIRL'S GUIDE TO MURDER SERIES

- Kill Joy (2021)

- A good girl's guide to murder (2019)

- Good girl, bad blood (2020)

- As Good As Dead (2021)
Tamsin Winter

**BOOKS**

- Being Miss Nobody (2017)

- Jemima Small Versus the Universe (2019)

**AGE RANGE**

Young Adult - younger teens

**WEBSITES AND MORE INFORMATION**

https://www.waterstones.com/books/search/term/tamsin+winter

https://www.tamsinwinter.com/

https://www.amazon.co.uk/Tamsin-Winter/e/B01N5R7AAH/ref=dp\_byline\_cont\_book\_1
Elizabeth Acevedo

## Basic Information

Elizabeth Acevedo is the New York Times bestselling author of the book *The Poet X* and is Dominican-American. She lives in Washington DC with her partner and identifies as Afro-Latina.

## Age Range

Elizabeth Acevedo writes for young adults.

## Books

* *The Poet X* (2018)
* *With The Fire On High* (2019)
* *Clap When You Land* (2020)
John Green

**BASIC INFORMATION**

John Michael Green is an American author and YouTube creator. He won the 2006 Printz Award for his first novel, *Looking for Alaska.* His fourth **solo** novel, *The Fault in Our Stars*, debuted at number one on *The New York Times* Best Seller list in January 2012.

**BIRTHDAY**

He was born on the 24th August 1977.

**AGE RANGE**

He writes for young adults.

**GENRE**

Contemporary.

**BOOKS**

**Young Adult**

* Looking for Alaska (2005)
* An Abundance of Katherines (2006)
* Let It Snow: Three Holiday Romances – with Maureen Johnson and Lauren Myracle (2008)
* Paper Towns (2008)
* Will Grayson, Will Grayson – with David Levithan (2010)
* The Fault in Our Stars (2012)
* Turtles All the Way Down (2017)

**Short stories**

* "The Approximate Cost of Loving Caroline" (2006)
* "The Great American Morp" (2007)
* "Freak the Geek", *Geektastic: Stories from the Nerd Herd* (2009)
* "Reasons", *What You Wish For* (2011)
* *Double on Call and Other Short Stories* (2012)

WEBSITES AND MORE INFORMATION

https://en.wikipedia.org/wiki/John\_Green\_(author)

http://www.johngreenbooks.com/

https://www.amazon.co.uk/John-Green/e/B001I9OQNE?ref=sr\_ntt\_srch\_lnk\_1&qid=1592831588&sr=8-1
Alex Armoredes

## Basic Information

Alex Armoredes was born in London, the city famous for Buckingham Palace, Big Ben (the bell), and copious amounts of rain. On a typical day Alex can be found either reading, solving puzzles, writing, or pondering the current state of existence. [1]

## Genres

Alex writes for both young adults and adults, as well as poetry. He tends to write short horror fiction.

## Books

* *Atom: A Collection of Poems (2022)*
* They Fade Away (2023)

## Websites and More Information

[1] https://www.goodreads.com/author/show/22870194.Alex\_Armoredes?ref=nav\_profile\_l

1. Alex Armoredes's Website: https://alexarmoredes.wordpress.com/
2. Alex Armoredes Twitter: https://twitter.com/armoredes