Geometry and the Circle Unit 2 > Lesson 1 of 6 A circle is an important shape in the field of geometry. Let's look at the definition of a circle and its parts. We will also examine the relationship between the circle and the plane. A circle is a shape with all points the same distance from its center. A circle is named by its center. Thus, the circle to the right is called circle A since its center is at point A. Some real world examples of a circle are a wheel, a dinner plate and (the surface of) a coin. The distance across a circle through the center is called the diameter. A real-world example of diameter is a 9-inch plate. The radius of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius. We can look at a pizza pie to find real-world examples of diameter and radius. Look at the pizza to the right which has been sliced into 8 equal parts through its center.

A radius is formed by making a straight cut from the center to a point on the circle. A straight cut made from a point on the circle, continuing through its center to another point on the circle, is a diameter. As you can see, a circle has many different radii and diameters, each passing through its center. A chord is a line segment that joins two points on a curve. In geometry, a chord is often used to describe a line segment joining two endpoints that lie on a circle. The circle to the right contains chord AB. If this circle was a pizza pie, you could cut off a piece of pizza along chord AB. By cutting along chord AB, you are cutting off a segment of pizza that includes this chord. A circle has many different chords. Some chords pass through the center and some do not. A chord that passes through the center is called a diameter. It turns out that a diameter of a circle is the longest chord of that circle since it passes through the center. A diameter satisfies the definition of a chord, however, a chord is not necessarily a diameter. This is because every diameter passes through the center of a circle, but some chords do not pass through the center.

Thus, it can be stated, every diameter is a chord, but not every chord is a diameter. Let's revisit the definition of a circle. A circle is the set of points that are equidistant from a special point in the plane. The special point is the center. In the circle to the right, the center is point A. Thus we have circle A. A plane is a flat surface that extends without end in all directions. In the diagram to the right, Plane P contains points A, B and C. Can you think of some real world objects that satisfy the definition of a plane? At this level of mathematics, that is difficult to do. Intuitively, a plane may be visualized as a flat infinite sheet of paper. The top of your desk and a chalkboard are objects which can be used to represent a plane, although they do not satisfy the definition above. A circle divides the plane into three parts: the points INSIDE the circle the points OUTSIDE the circle and the points ON the circle the points inside the circle the points outside the circle the points on the circle You can see an interactive demonstration of this by placing your mouse over the three items below.

A circle divides a plane into three parts: the points INSIDE the circle the points OUTSIDE the circle and the points ON the circle Example 1: The length of DB is 2.5 inches Summary: A circle is a shape with all points the same distance from its center. A circle is named by its center. The parts of a circle include a radius, diameter and a chord. All diameters are chords, but not all chords are diameters. A plane is a flat surface that extends without end in all directions. A circle divides the plane into three parts: The points inside the circle, the points outside the circle and the points on the circle. Exercises Directions: Refer to the diagram to answer each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button. 1. Which of the following is a chord, but not a diameter? circle page Parts of a Circle Circle - The set of all points in a plane that is the same distance from a given point called the center. Arc - Part of a circle; the curved line between two points on a circle.

If the circle is divided into two unequal parts, the smaller arc is called the minor arc and the larger arc is called the major arc. Center (of a Circle) - The point inside the circle that is the same distance from all points on the circle. Closed Curve - A curve that can be traced with the same starting and stopping points, and without crossing or retracing any part of the curve. A circle is an example of a closed curve. The circle is the set of points that lie on the blue curve. The points that lie inside this curve are not part of the circle (they form what is called a disk.) A circle is an example of a closed curve because you can begin at one point on the circle, trace it, and end up where you started without crossing or retracing any part of the circle. If the circle is divided into two unequal parts or arcs, the smaller arc is called the minor arc and the larger arc is called the major arc. The center of a circle is the same distance from all points on the circle. Point C is the center of this circle. This distance from the center of the circle to any point on the circumference is called the radius (indicated by r in the picture at the right).

Any line segment whose endpoints are the center and a point on the circle is also called the radius. Point E is not the center of this circle. Point E is not even part of the circle. Based on this definition, "same distance from all points", can a square have a center? Circles in Your World Sundials and quilts are often made of circles. The unusual floor plan of an apartment displays several arcs. Are they major or minor arcs? Check for Understanding Identify each of the parts of the circle at the right: a) segment AC c) the red curve, clockwise from D to C d) the blue curve, clockwise from C to D Choose each answer from the menus. a) External Links to Extend Basic Understanding Circles at Math.com : Read more about and practice naming parts of circles. Related Investigations on InterMath to Challenge Teachers Four dogs : Explore the paths of four dogs who each start at a different vertex of a square. Angles in a Circle : Investigate how the location of an angle's vertex affects its relationship to the measure(s) of the arc(s) it intercepts. Equation of a Circle : Find the equation of a circle based on its radius, center, and locus of points.

Optimizing Triangles using Radii : Find the triangles with maximum and minimum area if two sides of the triangle are determined by radii of the circle. Grazing for Mooey : How many square meters of grazing ground does Mooey the cow have? Top of Page Measurements Related to a Circle Area of a Circle - The amount of surface within a given boundary. Area is measured in squared units. The area of a circle can be found using A = pi*r*r (* means to multiply). Circumference - The distance around a circle. The circumference can be found using C = 2*pi*r (* means to multiply). Diameter - The distance across a circle through its center. The line segment that includes the center and whose endpoints lie on the circumference of the circle. Pi - The ratio of the circumference of a circle to its diameter ( = approximately 3.1416). Radius - The distance from the center of a circle to any point on the circumference. The line segment that has the center of the circle as one endpoint and a point on the circle as the other endpoint. The circumference (indicated in red) is like the perimeter around the circle, indicating the length or distance around the circle.

(It's similar, but not exactly alike. Perimeter deals with straight lines while circumference deals with curved lines.) As the number of sides of an inscribed regular polygon increases (see figures below), the perimeter of the regular polygon gets closer to the circumference of the circle. In other words, the perimeter of the regular octagon more closely approximates the circumference of the circle than does the perimeter of the regular pentagon or the equilateral triangle. The ratio of the circumference of any circle to its diameter is always a number that is close to 3.1416. This number is called pi and is written . Pi is an irrational number, which means it cannot be written as an exact decimal. We usually estimate pi with 3.14, 3.1416, or 22/7. The line segment that goes through the center of the circle and whose endpoints are on the circumference of the circle is called a diameter. The length of this line segment is also called the diameter. The line segment that has one endpoint as the center of the circle and the other endpoint on the circumference of the circle is called the radius of the circle. What do you think the relationship is between the radius and the diameter?

If you aren't sure, measure the diameter and radius of a circle. What do you notice? Measurements Related to a Circle in Your World What part of the bike wheel shown below is a radius? Why might you be interested in knowing about the circumference of the wheel? Check for Understanding Identify the following vocabulary terms based on a description: a) the distance across a circle, passing through the center b) the distance around a circle c) the distance from the circle to its center Choose each answer from the menus. a) Mrs. Glosser's Math Goodies--Circle Area : Read more about circles and practice calculating area. Related Investigations on InterMath to Challenge Teachers Tangential Circles : Compare the area and circumference of a circle inscribed in a region between two intersecting circles. Equal Areas : Construct a circle with five equal regions that do not intersect at the circle's center. Tangent Along the Diameter : Find the relationship between the area of a circle and the areas of the tangent circles along the circle's diameter. Changes in the Circumference : Explore how changing the diameter of a circle affects its circumference.

Area of a Sector : Determine the relationship between the area of a circle's sector and the radius and central angle of the circle, and between the arc length of a circle and the radius and central angle of the circle. Chord - A line segment that joins two points on the circumference of circle Examples of Chords The far right example shows the diameter as a chord of the circle. So the length of a chord is equal to or less than the diameter. Non-Examples of Chords Check for Understanding Identify all of the chords in the circle. Use two letters to represent each segment in your response. Enter each answer in the left column followed by pressing tab External Links to Extend Basic Understanding Circles at Math.com : Read more about chords and practice identifying them. Related Investigations on InterMath to Challenge Teachers Intersecting Chords : Determine the relationship between segments formed by the intersection of two chords of a circle. Perpendicular Bisectors : Describe the significance of the perpendicular bisectors of two segments that have endpoints on a given circle. Equidistant Chords : Determine the relationship between two segments that have endpoints on a circle if the segments are the same perpendicular distance away from the center of the circle.

Segments on a Secant : Find an equation that relates segments on a secant. Circumcircle - The circle that passes through all the vertices of a convex polygon. The figures below show circumcircles (in red) of regular polygons. The polygons do not have to be regular, but they do need to be convex, as the following illustrates. Related Investigations on InterMath to Challenge Teachers Inscribed Parallelogram : What must be true about a parallelogram that is inscribed in a circle? N-Sided Circle? : Compare the perimeters of polygons to the circumference of a circle. Concentric - Circles that have the same center. These concentric circles have the same center C. As the picture illustrates, the radii do not have to increase at a constant rate. In other words, the area between circles does not have to be the same "width". Circles at Math.com : Practice identifying radii in concentric circles. Related Investigations on InterMath to Challenge Teachers Target Areas : Determine the area of a target formed by concentric circles. Ready Fire Aim! : If three darts are thrown at a bulls-eye, what are the possible totals?

Impossible Scores : What is the highest score below 100 that is impossible to score on a given dart board? Semicircle - Half of a circle; a line containing the diameter of a circle separates the circle into two semicircles. In the figure, the line EF, which contains the diameter of circle A, divides the circle into two semi-circles, arc EGF and arc EHF. The degree measure of a semi-circle is 180 degrees, half that of a full circle which is 360 degrees. Circles Circles When you have finished this page, try the Circles Quiz . A circle is a plane figure with all points on the outside or circumference equally distant from the center of the figure. A line that is drawn from a point on one side of the circle to a point on the other side and passes through the center is the diameter of the circle. The diameter is a line segment, and thus is named by the letters that are at each endpoint. A line segment drawn from the center of the circle to a outside point of the circle is called a radius (plural radii). The radius is named by the letters at its endpoints.

A chord is a line segment that touches two points on the outside of the circle but doesn't pass through the center. The measure of the distance around the outside of a circle or the circle's perimeter is its circumference. A tangent is a line or line segment that touches the outside of the circle at a point, but does not pass through it in any way. A secant is a line that goes through or intersects a circle at two points and extends beyond the limits of the circle. An arc is a part of a circumference of the circle that has two endpoints on the circumference of the circle. All Elementary Mathematics - Study Guide - Geometry - Geometrical locus. Circle and circumference... Geometrical locus. Circle and circumference Geometrical locus. Midperpendicular. Angle bisector. Circumference. Circle. Arc. Secant. Chord. Diameter. Tangent line. Segment of a circle. Sector of a circle. Angles in a circle. Central angle. Inscribed angle. Circumscribed angle. Radian measure of angles. Round angle. Ratio of circumference length and diameter. Length of an arc. Huygens' formula. Relations between elements of a circle.

Geometrical locus ( or simply locus ) is a totality of all points, satisfying the certain given conditions. E x a m p l e 1. A midperpendicular of any segment is a locus, i.e. a totality of all points, equally removed from the bounds of the segment. Suppose that PO AB and AO = OB : Then, distances from any point P, lying on the midperpendicular PO, to bounds A and B of the segment AB are both equal to d . So, each point of a midperpendicular has the following property: it is removed from the bounds of the segment at equal distances. E x a m p l e 2. An angle bisector is a locus, that is a totality of all points, equally removed from the angle sides. E x a m p l e 3. A circumference is a locus, that is a totality of all points ( one of them - A ), equally removed from its center O. Circumference is a geometrical locus in a plane, that is a totality of all points, equally removed from its center.

Each of the equal segments, joining the center with any point of a circumference is called a radius and signed as r or R . A part of a plane inside of a circumference, is called a circle. A part of a circumference ( for instance, AmB, Fig.39 ) is called an arc of a circle. The straight line PQ, going through two points M and N of a circumference, is called a secant ( or transversal ). Its segment MN, lying inside of the circumference, is called a chord. A chord, going through a center of a circle ( for instance, BC, Fig.39 ), is called a diameter and signed as d or D . A diameter is the greatest chord of a circle and equal to two radii ( d = 2 r ). Tangent. Assume, that the secant PQ ( Fig.40 ) is going through points K and M of a circumference. Assume also, that point M is moving along the circumference, approaching the point K. Then the secant PQ will change its position, rotating around the point K. As approaching the point M to the point K, the secant PQ tends to some limit position AB.

The straight line AB is called a tangent line or simply a tangent to the circumference in the point K. The point K is called a point of tangency. A tangent line and a circumference have only one common point пїЅ a point of tangency. Properties of tangent. is an irrational number; its approximate value is 3.1415926пїЅ On the other hand, 2 is a round angle of a circumference, which in a degree measure is equal to 360 deg. In practice it often occurs, that both radius and angle of a circle are unknown. In this case, an arc length can be calculated by the approximate HuygensпїЅ formula: p 2l + ( 2l пїЅ L ) / 3 , where ( according to Fig.42 ): p пїЅ a length of the arc ACB; l пїЅ a length of the chord AC; L пїЅ a length of the chord AB. If an arc contains not more than 60 deg, a relative error of this formula is less than 0.5%. Relations between elements of a circle.

An inscribed angle ( ABC, Fig.45 ) is equal to a half of the central angle ( AOC, Fig.45 ), based on the same arc AmC. Therefore, all inscribed angles ( Fig.45 ), based on the same arc ( AmC, Fig.45 ), are equal. As a central angle contains the same quantity of degrees, as its arc ( AmC, Fig.45 ), then any inscribed angle is measured by a half of an arc, which is based on ( AmC in our case ). All inscribed angles, based on a semi-circle ( APB, AQB, пїЅ, Fig.46 ), are right angles ( Prove this, please ! ). An angle ( AOD, Fig.47 ), formed by two chords ( AB and CD ), is measured by a semi-sum of arcs, concluded between its sides: ( AnD + CmB ) / 2 . An angle ( AOD, Fig.48 ), formed by two secants ( AO and OD ), is measured by a semi-difference of arcs, concluded between its sides: ( AnD пїЅ BmC ) / 2 .

An angle ( DCB, Fig.49 ), formed by a tangent and a chord ( AB and CD ), is measured by a half of an arc, concluded inside of it: CmD / 2 .An angle ( BOC, Fig.50 ), formed by a tangent and a secant ( CO and BO ), is measured by a semi-difference of arcs, concluded between its sides: ( BmC пїЅ CnD ) / 2 . A circumscribed angle ( AOC, Fig.50 ), formed by the two tangents, (CO and AO), is measured by a semi-difference of arcs, concluded between its sides: ( ABC пїЅ CDA ) / 2 . Products of segments of chords ( AB and CD, Fig.51 or Fig.52 ), into which they are divided by an intersection point, are equal: AO пїЅ BO = CO пїЅ DO. A square of tangent line segment is equal to a product of a secant line segment by the secant line external part ( Fig.50 ): OA2 = OB пїЅ OD ( prove, please! ). This property may be considered as a particular case of Fig.52.

A chord ( AB, Fig.53 ), which is perpendicular to a diameter ( CD ), is divided into two in the intersection point O : AO = OB . ( Try to prove this ! ). Circles and using a Compass Year 8 Interactive Maths - Second Edition Circles and using a Compass In this section, we will consider the definition of a circle, lines in a circle and parts of a circle. Circle A circle is a set of points in a plane that are the same distance from a fixed point (called the centre). These set of points form the perimeter of the circle. The radius is the distance from the centre of the circle to any point on its perimeter. The circumference of a circle is the perimeter of the circle. These parts of a circle are indicated in the accompanying diagram. Note: The plural of radius is radii. Lines in a Circle The name of a line in a circle depends on its position in the circle. A secant is a line that passes through any two points on a circle. A chord is a line that joins two points on the circumference of a circle. The diameter is a chord that passes through the centre of a circle. A tangent is a line that touches the circle at only one point.

Parts of a Circle An arc is a part of the circumference . A sector is the part of a circle between two radii . A segment is the part of a circle that is between a chord and the circumference . A semicircle is a half of a circle. Compass A compass is an instrument used to draw circles or the parts of circles called arcs . It consists of two movable arms hinged together where one arm has a pointed end and the other arm holds a pencil. Note that a compass is also called a pair of compasses. To draw a circle (or arc) with a compass: make sure that the hinge at the top of the compass is tightened so that it does not slip tighten the hold for the pencil so it also does not slip align the pencil lead with the compass's needle press down the needle and turn the knob at the top of the compass to draw a circle (or arc ) Example 2 Use a compass to draw a circle of radius 4 cm. Solution: Step 1: Use a ruler to set the distance from the point of the compass to the pencil's lead at 4 cm. Step 2: Place the point of the compass at the centre of the circle.

Step 3: Draw the circle by turning the compass through 360ï¿½. Activity 10.1 1. Use a compass to draw a circle of radius 5 cm. 2. Use a compass to draw a circle of diameter 12 cm. 3a. Use a compass to draw a circle of radius 4.5 cm. 3b. Draw the diameter of the circle; and use a ruler to measure the length of the diameter. 3c. Write an equation to represent the relation between the radius, r, and the diameter, d. 4a. Use a compass to draw a circle of radius 5.5 cm. 4b. Draw a diameter and label it PQ. 4c. Draw a triangle PQR where R is on the semicircle. 4d. Use a protractor to measure the size of angle PRQ. 5a. Use a compass to draw a circle of radius 6.5 cm. 5b. Draw a diameter and label it PQ. 5c. Draw a triangle PQR where R is on the semicircle. 5d. Use a protractor to measure the size of angle PRQ. 6a.

Use a compass to draw a circle of radius 7.5 cm. 6b. Draw a diameter and label it PQ. 6c. Draw a triangle PQR where R is on the semicircle. 6d. Use a protractor to measure the size of angle PRQ. 7. Use the results of questions 4, 5 and 6 to complete the following statements: a. The size of the angle on the diameter of a circle with a vertex on the circle is ï¿½ b. If a triangle is drawn in a semicircle using the diameter as an edge, the angle touching the curved part of the triangle is ï¿½ Key Terms Circle Geometry Circle Geometry Measurement and Geometry : Module 26 Years : 9-10 return to top Assumed knowledge Introductory plane geometry involving points and lines, parallel lines and transversals, angle sums of triangles and quadrilaterals, and general angle-chasing. Experience with a logical argument in geometry written as a sequence of steps, each justified by a reason. Ruler-and-compasses constructions. The four standard congruence tests and their application to proving properties of and tests for special triangles and quadrilaterals. The four standard similarity tests and their application.

Trigonometry with triangles. return to top Motivation Most geometry so far has involved triangles and quadrilaterals, which are formed by intervals on lines, and we turn now to the geometry of circles. Lines and circles are the most elementary figures of geometry − a line is the locus of a point moving in a constant direction, and a circle is the locus of a point moving at a constant distance from some fixed point − and all our constructions are done by drawing lines with a straight edge and circles with compasses. Tangents are introduced in this module, and later tangents become the basis of differentiation in calculus. The theorems of circle geometry are not intuitively obvious to the student, in fact most people are quite surprised by the results when they first see them. They clearly need to be proven carefully, and the cleverness of the methods of proof developed in earlier modules is clearly displayed in this module. The logic becomes more involved − division into cases is often required, and results from different parts of previous geometry modules are often brought together within the one proof. Students traditionally learn a greater respect and appreciation of the methods of mathematics from their study of this imaginative geometric material.

The theoretical importance of circles is reflected in the amazing number and variety of situations in science where circles are used to model physical phenomena. Circles are the first approximation to the orbits of planets and of their moons, to the movement of electrons in an atom, to the motion of a vehicle around a curve in the road, and to the shapes of cyclones and galaxies. Spheres and cylinders are the first approximation of the shape of planets and stars, of the trunks of trees, of an exploding fireball, and of a drop of water, and of manufactured objects such as wires, pipes, ball-bearings, balloons, pies and wheels. return to top Radii and chords We begin by recapitulating the definition of a circle and the terminology used for circles. Throughout this module, all geometry is assumed to be within a fixed plane. A circle is the set of all points in the plane that are a fixed distance (the radius) from a fixed point (the centre). Any interval joining a point on the circle to the centre is called a radius. By the definition of a circle, any two radii have the same length. Notice that the word ‘radius’ is being used to refer both to these intervals and to the common length of these intervals.

An interval joining two points on the circle is called a chord. A chord that passes through the centre is called a diameter. Since a diameter consists of two radii joined at their endpoints, every diameter has length equal to twice the radius. The word ‘diameter’ is use to refer both to these intervals and to their common length. A line that cuts a circle at two distinct points is called a secant. Thus a chord is the interval that the circle cuts off a secant, and a diameter is the interval cut off by a secant passing through the centre of a circle centre. Symmetries of a circle Circles have an abundance of symmetries: A circle has every possible rotation symmetry about its centre, in that every rotation of the circle about its centre rotates the circle onto itself. If AOB is a diameter of a circle with centre O, then the reflection in the line AOB reflects the circle onto itself. Thus every diameter of the circle is an axis of symmetry. As a result of these symmetries, any point P on a circle can be moved to any other point Q on the circle. This can be done by a rotation through the angle θ = POQ about the centre.

It can also be done by a reflection in the diameter AOB bisecting POQ. Thus every point on a circle is essentially the same as every other point on the circle − no other figure in the plane has this property except for lines. EXERCISE 1 Which of the transformations in part a map a particular point P on to another particular point Q on . Congruence and similarity of circles Any two circles with the same radius are congruent− if one circle is moved so that its centre coincides with the centre of the other circle, then it follows from the definition that the two circles will coincide. More generally, any two circles are similar − move one circle so that its centre coincides with the centre of the other circle, then apply an appropriate enlargement so that it coincides exactly with the second circle. A circle forms a curve with a definite length, called the circumference, and it encloses a definite area. The similarity of any two circles is the basis of the definition of π, the ratio of the circumference and the diameter of any circle.

We saw in the module, The Circles that if a circle has radius r, then circumference of the circle = 2πr and area of the circle = πr2 Radii and chords Let AB be a chord of a circle not passing through its centre O. The chord and the two equal radii OA and BO form an isosceles triangle whose base is the chord. The angle AOB is called the angle at the centre subtended by the chord. In the module, Rhombuses, Kites and Trapezia we discussed the axis of symmetry of an isosceles triangle. Translating that result into the language of circles: Theorem Let AB be a chord of a circle with centre O. The following three lines coincide: The bisector of the angle AOB at the centre subtended by the chord. The interval joining O and the midpoint of the chord AB. The perpendicular bisector of the chord AB. Trigonometry and chords Constructions with radii and chords give plenty of opportunity for using trigonometry.

EXERCISE 2 OM = cos θ This exercise shows that sine can be regarded as the length of the semichord AM in a circle of radius 1, and cosine as the perpendicular distance of the chord from the centre. Until modern times, tables of sines were compiled as tables of chords or semichords, and the name ‘sine’ is conjectured to have come in a complicated and confused way from the Indian word for semichord. Arcs and sectors Let A and B be two different points on a circle with centre O. These two points divide the circle into two opposite arcs. If the chord AB is a diameter, then the two arcs are called semicircles. Otherwise, one arc is longer than the other − the longer arc is called the major arc AB and the shorter arc is called the minor arc AB. Now join the radii OA and OB. The reflex angle AOB is called the angle subtended by the major arc AB. The non-reflex angle AOB is called the angle subtended by the minor arc AB, and is also the angle subtended by the chord AB. The two radii divide the circle into two sectors, called correspondingly the major sector OAB and the minor sector OAB.

It is no surprise that equal chords and equal arcs both subtend equal angles at the centre of a fixed circle. The result for chords can be proven using congruent triangles, but congruent triangles cannot be used for arcs because they are not straight lines, so we need to identify the transformation involved. Theorem and hence are equal. Segments A chord AB of a circle divides the circle into two segments. If AB is a diameter, the two congruent segments are called semicircles − the word ‘semicircle’ is thus used both for the semicircular arc, and for the segment enclosed by the arc and the diameter. Otherwise, the two segments are called a major segment and a minor segment. The word ‘subtend’ The word ‘subtend’ literally means ‘holds under’, and is often used in geometry to describe an angle. Suppose that we have an interval or arc AB and a point P not on AB. Join the intervals AP and BP to form the angle APB. We say that the angle APB is the angle subtended by the interval or arc AB at the point P.

return to top Angles in a semicircle Let AOB be a diameter of a circle with centre O, and let P be any other point on the circle. The angle APB subtended at P by the diameter AB is called an angle in a semicircle. This angle is always a right angle − a fact that surprises most people when they see the result for the first time. Theorem An angle in a semicircle is a right angle. Proof Let AB be a diameter of a circle with centre O, and let P be any other point on the circle. Join the radius PO, and let α = A and β = The triangles AOP and BOP are isosceles because all radii are equal, so APO = α and BPO = β (base angles of isosceles triangles AOP and BOP). Hence α + β + (α + β ) = 180° (angle sum of APB) This famous theorem is traditionally ascribed to the Greek mathematician Thales, the first known Greek mathematician. EXERCISE 3 Thales’ theorem can also be proven using the following test for a rectangle developed in the module, Parallelograms and Rectangles .

A quadrilateral whose diagonals are equal and bisect each other is a rectangle. Join the radius PO and produce it to a diameter POQ, then join up the quadrilateral APBQ. Explain why APB is a right angle. Constructing a right angle at the endpoint of an interval Thales’ theorem gives a quick way to construct a right angle at the endpoint of an interval AX. 1 PAQ = 90°. The converse theorem The angle in a semicircle theorem has a straightforward converse that is best expressed as a property of a right-angled triangle: Theorem The circle whose diameter is the hypotenuse of a right-angled triangle passes through all three vertices of the triangle. Proof ABC be right-angled at C, and let M be the midpoint of the hypotenuse AB. We need to prove that MC = MA = MB. Complete the rectangle ACBR. Because ACBR is a rectangle, its diagonals bisect each other and are equal. Hence M is the midpoint of the other diagonal CR, and AM = BM = CM = RM . This is an excellent example of the way ideas in geometry fit together − a significant theorem about circles has been proven using a property of rectangles.

Two practical situations to illustrate the converse theorem A set of points in the plane is often called a locus. The term is used particularly when the set of points is the curve traced out by a moving point. For example, a circle can be defined as the locus of a point that moves so that its distance from some fixed point is constant. The two examples below use the converse of the angle in a semicircle theorem to describe a locus. EXERCISE 4 return to top Angles at the centre and circumference The angle-in-a-semicircle theorem can be generalised considerably. In each diagram below, AB is an arc of a circle with centre O, and P is a point on the opposite arc. The arc AB subtends the angle AOB at the centre. The arc also subtends the angle APB, called an angle at the circumference subtended by the arc AB. In the middle diagram, where the arc is a semicircle, the angle at the centre is a straight angle, and by the previous theorem, the angle at the circumference is a right angle − exactly half.

We shall show that this relationship holds also for the other two cases, when the arc is a minor arc (left-hand diagram) or a major arc (right-hand diagram). The proof uses isosceles triangles in a similar way to the proof of Thales’ theorem. Theorem An angle at the circumference of a circle is half the angle at the centre subtended by the same arc. Proof Let AB be an arc of a circle with centre O, and let P be any point on the opposite arc. We need to prove AOB = 2 APB. The proof divides into three cases, depending on whether: Case 1: O lies inside ABP Case 2: O lies on APB Case 3: O lies outside APB. Case 1: Join PO and produce PO to Q. Then OA = OB = OP (radii), so we have two isosceles triangles OAP and OAQ. Let As the horse moves from position A to position B, the horse moves an angle AOB = 2θ about the centre O of the track.

The angle at the circumference is half the angle at the centre, so APB = θ which means that the punter’s binoculars rotate by an angle θ the horse moves from A to B. Hence the punter is rotating his binoculars at a constant rate that is half the rate at which the horse is rotating about the centre. Angles subtended by the same arc In the diagram to the right, the two angles APB and are subtended by the same (minor) arc AB. Each angle is half the angle AOB at the centre subtended by the same arc, so This corollary of the previous theorem is a particularly significant result about angles in circles: Theorem Two angles at the circumference subtended by the same arc are equal. Thales’ theorem is a special case of this theorem. Some alternative terminology The last two theorems are often expressed in slightly different language, and some explanation is needed to avoid confusion. 1 An angle subtended by an arc is often said to be standing on the arc. With this terminology, the two theorems become: - An angle at the circumference of a circle is half the angle at the centre standing on the same arc.

- Two angles at the circumference standing on the same arc are equal. In the context of these two theorems, it is best to avoid the phrases ‘standing on a chord AB’ and ‘subtended by a chord AB′, because we need to distinguish between angles subtended by the major arc AB and angles subtended by the minor arc AB. 2 In the upper diagram to the right, the angle APB is called an angle in the (major) segment. Notice that it actually stands on the minor arc AB, which can be confusing. We have already used this terminology before in speaking about an ‘angle in a semicircle’. With this terminology, the last theorem becomes: Two angles in the same segment of a circle are equal. The situation is illustrated in the lower diagram to the right. Extension − The orthocentre of a triangle An altitude of a triangle is a perpendicular from any of the three vertices to the opposite side, produced if necessary. The two cases are illustrated in the diagrams below. There are three altitudes in a triangle. The following theorem proves that they concurrent at a point called the orthocentre H of the triangle. It is surprising that circles can be used to prove the concurrence of the altitudes.

Theorem The altitudes of a triangle are concurrent. return to top Cyclic quadrilaterals In the module, Congruence , we showed how to draw the circumcircle through the vertices of any triangle. To do this, we showed that the perpendicular bisectors of its three sides are concurrent, and that their intersection, called the circumcentre of the triangle, is equidistant from each vertex. No other circle passes through these three vertices. If we tried to take as centre a point P other than the circumcentre, then P would not lie on one of the perpendicular bisectors, so it could not be equidistant from the three vertices. When there are four points, we can always draw a circle through any three of them (provided they are not collinear), but only in special cases will that circle pass through the fourth point. A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle. This is the last type of special quadrilateral that we shall consider. Constructing the circumcircle of a cyclic quadrilateral Suppose that we are given a quadrilateral that is known to be cyclic, but whose circumcentre is not shown (perhaps it has been rubbed out).

The circumcentre of the quadrilateral is the circumcentre of the triangle formed by any three of its vertices, so the construction to the right will find its circumcentre. The opposite angles of a cyclic quadrilateral The distinctive property of a cyclic quadrilateral is that its opposite angles are supplementary. The following proof uses the theorem that an angle at the circumference is half the angle at the centre standing on the same arc. Theorem The opposite angles of a cyclic quadrilateral are supplementary. Proof Let ABCD be a cyclic quadrilateral with O the centre of the circle. Join the radii OB and OD. Let α and g be the angles at the centre, as shown on the diagram. Then Extension − A test for a cyclic quadrilateral The property of a cyclic quadrilateral proven earlier, that its opposite angles are supplementary, is also a test for a quadrilateral to be cyclic. That is the converse is true. This theorem completes the structure that we have been following − for each special quadrilateral, we establish its distinctive properties, and then establish tests for it. The proof uses ‘proof by contradiction’, and is thus a little more difficult than other Year 10 material. If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

Proof Let ABCD be a quadrilateral with A + C = 180°. Construct the circle through A, B and D, and suppose, by way of contradiction, that the circle does not pass through C. Let DC, produced if necessary, meet the circle again at X, and join XB. Then BXD are supplementary because ABXD is a cyclic quadrilateral, so the angles DCB are equal, so XB || CB Hence XB and CB are the same line, so C and X coincide, that is the circle does pass through C. EXERCISE 11 Prove the following alternative form of the above theorem: If an exterior angle of a quadrilateral equals the opposite interior angle, then the quadrilateral is cyclic. EXERCISE 12 Prove that the trapezium is cyclic. Extension − The sine rule and circle geometry The sine rule states that for any triangle ABC, the ratio of any side over the sine of its opposite angle is a constant, Each term is the ratio of a length over a pure number, so their common value seems to be a length. Thus it reasonable to ask, what is this common length?

The answer is a surprise − the common length is the diameter of the circumcircle through the vertices of the triangle. The proof of this result provides a proof of the sine rule that is independent of the proof given in the module, Complete the proof in Case 3. Tangents to circles A tangent to a circle is a line that meets the circle at just one point. The diagram below shows that given a line and a circle, can arise three possibilities: The line may be a secant, cutting the circle at two points. The line may be a tangent, touching the circle at just one point. The line may miss the circle entirely. The words ‘secant’ and ‘tangent’ are from Latin − ‘secant’ means ‘cutting’ (as in ‘cross-section’), and ‘tangent’ means ‘touching’ (as in ‘tango’). The point where a tangent touches a circle is called a point of contact. It is not immediately obvious how to draw a tangent at a particular point on a circle, or even whether there may be more than one tangent at that point. The following theorem makes the situation clear, and uses Pythagoras’ theorem in its proof.

Theorem Let T be a point on a circle with centre O. a Every point on the tangent, except for T itself, lies outside the circle. Proof First we prove parts a and c. Let be the line through T perpendicular to the radius OT. Let P be any other point on , and join the interval OP. Then using Pythagoras’ theorem in OTP, OP2 = OT2 + PT2; which is greater than OT2, so OP is greater than the radius OT. Hence P lies outside the circle, and not on it. This proves that the line is a tangent, because it meets the circle only at T. It also proves that every point on , except for T, lies outside the circle. It remains to prove part b, that there is no other tangent to the circle at T. Let t be a tangent at T, and suppose, by way of contradiction, that t were not perpendicular to OT. Let M be the foot of the perpendicular from O to t, and construct the point U on t,on the other side of M, so that UM = MT.

Then using Pythagoras’ theorem in OMT and OMU, OT2 = OM2 + MT2 = OM2 + MU2 = OU2, So OU = OT. Hence U also lies on the circle, contradicting the fact that t is a tangent. Construction − Tangents from an external point Using this radius and tangent theorem, and the angle in a semi circle theorem, we can now construct tangents to a circle with centre O from a point P outside the circle. Join OP and construct the midpoint M of OP . 2 Draw the circle with centre M through O and P, and let it meet the circle at T and U. The angles PTO and PUO are right angles, because they are angles in a semicircle. Hence the lines PT and PU are tangents, because they are perpendicular to the radii OT and OU, respectively. Tangents from an external point have equal length It is also a simple consequence of the radius-and-tangent theorem that the two tangents PT and PU have equal length. Notice that here, and elsewhere, we are using the word ‘tangent’ in a second sense, to mean not the whole line, but just the interval from an external point to the point of contact.

Tangents to a circle from an external point have equal length. EXERCISE 14 Prove this result using either congruence or Pythagoras’ theorem. Tangents and trigonometry The right angle formed by a radius and tangent gives further opportunities for simple trigonometry. The following exercise shows how the names ‘tangent’ and ‘secant’, and their abbreviations tan θ and sec θ, came to be used in trigonometry. In a circle of radius 1, the length of a tangent subtending an angle θ at the centre is tan θ , and the length of the secant from the external point to the centre is sec θ . EXERCISE 15 . Quadrilaterals with incircles The following exercise involves quadrilaterals within which an incircle can be drawn tangent to all four sides. These quadrilaterals form yet another class of special quadrilaterals. EXERCISE 16 The sides of a quadrilateral are tangent to a circle drawn inside it. Show that the sums of opposite sides of the quadrilateral are equal. Common tangents and touching circles A line that is tangent to two circles is called a common tangent to the circles.

When the points of contact are distinct, there are two cases, as in the diagrams below. The two circles lie on the same side of a direct common tangent, and lie on opposite sides of an indirect common tangent. Direct common tangent Indirect common tangent Two circles are said to touch at a common point T if there is a common tangent to both circles at the point T. As in the diagram below, the circles touch externally when they are on opposite sides of the common tangent, and touch internally when they are on the same side of the common tangent. Provided that they are distinct, touching circles have only the one point in common. EXERCISE 17 are next to each other without touching? b Prove that the intervals between the points of contact on two indirect common tangents have equal length. c Prove that the intervals between the points of contact on two direct common tangents have equal length. Extension - The incentre of a triangle We are now in a position to prove a wonderful theorem on the angle bisectors of a triangle. These three bisectors are concurrent, and their point of intersection is called the incentre of the triangle.

The incentre is the centre of the incircle tangent to all three sides of the triangle, as in the diagram to the right. Theorem The angle bisectors of a triangle are concurrent, and the resulting incentre is the centre of the incircle, that is tangent to all three sides. Proof We have to prove IC bisects BCD ABC , construct the angle bisectors of A and B, and let I be their point of intersection. Join IC, and let α = BAI = MCI. d Hence complete the proof that the angle bisectors meet at I, and that I is the incentre. Summary - The four centres of a triangle This completes the development of the four best-known centres of a triangle. The results, and the associated terminology and notation, are summarised here for reference. The perpendicular bisectors of the three sides of a triangle are concurrent at the circumcentre O, which is the centre of the circumcircle through all three vertices. This was proven in the module, Congruence . The angle bisectors of the three angles of a triangle are concurrent at the incentre I, which is the centre of the incircle that is tangent to all three sides. This was proven above.

A median of a triangle is an interval joining a vertex to the midpoint of an opposite side. The medians of a triangle are concurrent at the centroid G, which divides each median in the ratio 2 : 1. This was proven in the module, Scale Drawings and Similarity . An altitude of a triangle is a perpendicular from a vertex to the opposite side, produced if necessary. The altitudes of a triangle, also produced if necessary, are concurrent at the orthocentre H. This was proven earlier in the present module as an extension in the section on angles at the centre and circumference . The left-hand diagram below shows two angles P and Q lying in the same segment of a circle − we have proven that these two angles are equal. In the next two diagrams, the angle BQU remains equal to P as the point Q moves around the arc closer and closer to A. In the last diagram, Q coincides with A, and AU is a tangent. It seems reasonable from these diagrams that in the limiting case, the angle BAU will still equal P − this is what the alternate segment theorem says. Theorem An angle between a chord and a tangent is equal to any angle in the alternate segment.

Here BAU is ‘an angle between a chord and a tangent’. The word ‘alternate’ means ‘other’ − the chord AB divides the circle into two segments, and the alternate segment is the segment on the left containing the angle P. We have already proven that all the angles in this left-hand segment are equal. There are two equally satisfactory proofs of this theorem. One is written out below and the other is left as an exercise. Proof Let AB be a chord of a circle and let BU be a tangent at B. Let P be a point on the arc that is not within the arms of ABU , and let There are three cases, depending on whether: Case 1: The centre O lies on the chord AB. Case 2: The centre O lies outside the arms of ABU . Case 3: The centre O lies within the arms of ABU. In each case, we need to prove that ABU = α return to top Similarity and circles The final theorems in this module combine similarity with circle geometry to produce three theorems about intersecting chords, intersecting secants, and the square on a tangent. Intersecting chords The first theorem deals with chords that intersect within the circle.

In this situation, each chord cuts the other into two sub-intervals called intercepts. It is an amazing consequence of similar triangles that, in this situation, the products of the intercepts on each chord are equal. That is, in the diagram to the right, AM × MB = PM × MQ. Theorem When two chords intersect within a circle, the products of the intercepts are equal. Proof Let AB and PQ be chords intersecting at M. Join AP and BQ. In the triangles APM and QBM: PAM = BQM (angles on the same arc PB) APM = QBM (angles on the same arc AQ) so (matching sides of similar triangles) so AM × BM = PM × QM. The very last step is particularly interesting. It converts the equality of two ratios of lengths to the equality of two products of lengths. This is a common procedure when working with similarity. The sine rule and similarity Many problems involving similarity can be handled using the sine rule. The exercise below gives an alternative proof of the intersecting chord theorem using the sine rule to deal directly with the ratio of two sides of the triangles. The remaining two theorems of this section also have alternative proofs using the sine rule.

EXERCISE 21 Secants from an external point Now suppose that the chords do not meet within the circle. If they meet on the circle, the identity above holds trivially, and if they are parallel, there is nothing to say. Otherwise, we can produce the chords until they intersect outside the circle, and an analogous theorem applies. We are now dealing with secants from an external point. When a secant through an external point M meets a circle at two points A and B, the lengths AM and BM are called the intercepts of the secant from the external point, and as before AM × MB = PM × MQ. Be careful here, the chord AB is not one of the two intercepts. With this definition of intercept, the previous theorem can now be stated as follows. Its proof is word for word almost the same, apart from the reasons: Theorem The product of the intercepts on two secants from an external point are equal. Proof Let ABM and PQM be two secants from an external point M, as shown. Join AP and BQ.

Tangent and secant from an external point Now imagine the secant MPQ in the previous diagrams rotating until it becomes a tangent at a point T on the circle. As the secant rotates, the length of each intercept PM and QM gets closer to the length of the tangent TM to the circle from M, so the product PM × QM gets closer to the square TM2 of the tangent from M. This is a proof using limits. When we draw a secant and a tangent from M, we have seen that the product AM × BM equals the square TM2 of the tangent. The geometric proof is similar to the previous two proofs, but it does require the alternate segment theorem to establish the similarity. Theorem The product of the intercepts on a secant from an external point equals the square of the tangent from that point. Proof Let ABM be a secant, and TM a tangent, from an external point M, as shown. We need to prove that AM × BM = TM2. Join AT and BT.

EXERCISE 24 return to top converse of the circle theorems The circle theorems proven in this module all have dramatic and important converse theorems, which are tests for points to lie on a circle. The proofs of these converses, and their applications, are usually regarded as inappropriate for Years 9−10, apart from the converse of the angle in a semicircle theorem, which was developed within the module. They are so closely related to the material in this module, however, that they have been fully developed in an Appendix. Coordinate geometry Tangents to circles We defined a tangent to a circle as a line with one point in common with the circle. This definition can be used in coordinate geometry using simultaneous equations. For example, the diagram to the right shows the line x + y = 2 and the circle x2 + y2 = 2. Substituting the equation of the line into the equation of the circle gives x2 + (2 − x)2 (x − 1)2 = 0 and the only solution is thus x = 1.

Hence the line and the circle have only the single point of intersection T(1, 1), proving that the line is a tangent to the circle. Similarly, the dotted line x + y = 1 is a secant, intersecting the circle in two points, and the dotted line x + y = 3 does not intersect the circle at all. Tangents to parabolas This simultaneous equations approach to tangents can be generalised to other curves defined by algebraic equations. For example, the line y = 2x − 1 is a tangent to the graph of the parabola of y = x2 at the point P(1, 1), because solving the equations simultaneously gives x2 = 2x − 1 return to top Calculus For graphs defined by cubics and higher powers, however, the definition of ‘tangent’ has to be adapted. The first diagram to the right shows that we would think of as a ‘tangent’ at a point P can cross the curve again at some other point, Q. The following diagram shows that even with a quadratic graph, our current definition of ‘tangent’ would mean that every vertical line would be a tangent to the parabola!

Clearly we need to change the requirement of a single point of intersection, and instead develop some idea about a tangent being a straight line that ‘approximates a curve’ in the neighbourhood of the point of contact. These considerations lead naturally to the well-known limiting process. To construct a tangent at a point P on a general curve, we construct the secant through P and another point Q on the curve, and then move the point Q closer and closer to P. This is the traditional beginning of calculus at school. Applications in motion and rates of change When a stone on a string is whirled around in a circle, then suddenly let go, it flies off at a tangent to the circle (ignoring the subsequent fall to the ground). We can interpret this situation by saying that the tangent describes the direction in which the stone was travelling at the instant when it was released. This leads to the concept of a vector which has both magnitude and direction representing the velocity particle. The study of motion begins with motion in a straight line, that is, in one dimension.

When motion in two dimensions is first considered, circular paths and parabolic paths are the first paths to be considered, because they are reasonably straightforward to describe mathematically, and they arise in so many practical situations. A rate of change is also described by a tangent. This time the tangent does not have any physical significance, but it is the gradient of the tangent that gives the instantaneous rate of change. For example, when boiling water is removed from the stove and cools, the temperature−time graph looks something like the graph to the right. At any time t, the rate at which the water is cooling is given by the gradient of the tangent at the corresponding point on the curve. Vectors and complex numbers Problems in complex numbers often require locating a set of complex numbers on the complex plane. The following example requires some knowledge of vectors and circle geometry. EXAMPLE is not defined. So suppose that z is neither 1 nor −1. The condition on z is that z − 1 is ki times z + 1, where k is real. We know that multiplying by ki rotates the direction of a complex number by 90° or by −90°. That is, the vector z − 1 is perpendicular to the vector z + 1.

We know that z − 1 is a vector with tail at 1 and head at z, and that z + 1 is a vector with tail at −1 and head at z. Let A, B and P represent the points −1, 1 and z. Then the condition becomes APB is a right angle. Thus the locus of z is a circle with diameter AB, that is a circle of radius 1 and centre 0, excluding −1. Many problems similar to this involve not just the theorems developed in the module, but their converses as well, as developed in the Appendix to this module. return to top History and applications Greek geometry was based on the constructions of straight lines and circles, using a straight edge and compasses, which naturally gave circles a central place in their geometry. All the theorems developed in the Content and Appendix of this module were developed by the Greeks, and appear in Euclid’s Elements. In coordinate geometry, developed later by Descartes in the 17th century, horizontal and vertical lengths are measured against the two axes, and diagonal lengths are related to them using Pythagoras’ theorem.

The resulting connection between circles and Pythagoras’ theorem is seen in the equation of a circle. The square of the distance of a point P(x, y) from the origin is x2 + y2 by Pythagoras’ theorem, which means that the equation of the circle with radius a and centre the origin is x2 + y2 = a2. A circle is a simple closed curve with an inside and an outside, a property that it shares with triangles and quadrilaterals. In three dimensions, spheres, cubes and toruses (doughnuts) have an inside and an outside, but a torus is clearly connected in a different way from a sphere. The subject called topology, begun by Euler and developed extensively in the 20th century, begins with such observations. Greek astronomy made great use of circles and spheres. They knew that the Earth was round, and were able to calculate its circumference with reasonable accuracy. Ptolemy described the heavenly bodies in terms of concentric spheres on which the Moon, the planets, the Sun and the stars were embedded.

When astronomy was reconstructed in the 16th century by Copernicus, the heavenly spheres disappeared, but he still used circles for orbits, with the centre moved from the Earth to the Sun. Ellipses and other refinements of the orbits were soon introduced into this basic model of circles by Kepler, who empirically found three laws of motion for planets. At the end of the 17th century Newton used calculus, his laws of motion and the universal law of gravitation to derive Kepler’s laws. John Dalton reconstructed chemistry at the start of the 19th century on the basis of atoms, which he regarded as tiny spheres, and in the 20th century, models of circular orbits and spherical shells were originally used to describe the motion of electrons around the spherical nucleus. Thus circles and their geometry have always remained at the heart of theories about the microscopic world of atoms and theories about the cosmos and the universe. Geometry continues to play a central role in modern mathematics, but its concepts, including many generalisations of circles, have become increasingly abstract.

For example, spheres in higher dimensional space came to notice in 1965, when John Leech and John Conway made a spectacular contribution to modern algebra by studying an extremely close packing of spheres in 24-dimensional space. On the other hand, classical Euclidean geometry in the form presented in this module has nevertheless advanced in modern times − here are three results obtained in recent centuries. In 1765, Euler discovered that: The orthocentre, the circumcentre, and the centroid of any triangle are collinear. The centroid divides the interval joining the circumcentre and the orthocentre in the ratio 2:1. The line joining these three centres is now called the Euler line. The proof is reasonably straightforward, and is presented in the following exercise. EXERCISE 26 In the diagram to the right, G and O are respectively the centroid and the circumcentre of ABC. The interval OG is produced to X so that OG : GX = 1 : 2. We shall prove that X is the orthocentre of ABC by proving that it lies on each of the three altitudes of the triangle. Join AX. Let M be the midpoint of BC. Join the median AGM, and the perpendicular OM.

a In the early 19th century, Poncelet, Feuerbach and others showed that in any triangle, the following nine points are cyclic: the midpoint of each side of the triangle, the foot of each altitude, the midpoint of the interval joining each vertex of the triangle to the orthocentre. That is, these nine points lie on a circle. He also showed that the centre of this nine-point circle lies on the Euler line, and is the midpoint of the interval joining the circumcentre to the orthocentre. Again, the proof is straightforward enough to present here as a structured exercise, although proving that the centre of the nine-point circle is the midpoint of OH is rather fiddly. EXERCISE 27 The diagram to the right shows ABC with the altitudes AU, BV and CW meeting at the orthocenter H. The points E, F and G are the respective midpoints of AH, BH and CH. The points P, Q and R are the midpoints of the sides BC, CA and AB. a Explain why N lies on OH, and is thus its midpoint.

An enlargement transformation associated with Euler’s line and the nine-point circle Here is a rather dramatic proof using a single enlargement that establishes the existence of the Euler line and the positions on it of the centroid G, the circumcentre O, the orthocentre H, and the nine-point centre N. The notation is the same as that used in exercises 25 and 26. In this construction, all that is used about the nine-point circle point is that it is the circle through P, Q and R. The fact that the other six points lie on it would be proven afterwards. The medians trisect each other, so AG : GP = BG : GQ = CG : GR = 2 : 1. Consider the enlargement with centre G and enlargement factor − . This enlargement fixes G, and maps A to P, B to Q and C to R, so the circumcircle of ABC is mapped to the circumcircle of PQR. Thus the circumcentre O is mapped to the nine-point centre N, and G thus divides NO in the ratio 1 : 2.

The orthocentre H of ABC is mapped to the orthocentre of PQR, and the orthocentre of PQR is the circumcentre O of ABC, because the perpendicular bisectors of the sides of ABC are the altitudes of PQR, so G divides HO in the ratio 2 : 1. Thus the four points H, N, G and O are collinear, with HN : NG : GO = 3 : 1 : 2. It also follows that the nine-point circle has half the radius of the circumcircle. return to top Morley’s trisector theorem In 1899, the American mathematician Frank Morley discovered an amazing equilateral triangle that is formed inside every triangle. When each angle of a triangle is trisected, the points of intersection of trisectors of adjacent vertices form an equilateral triangle. The result has many proofs by similar triangles, and we refer the reader particularly to John Conway’s proof and Bollobas’ version . The result can also be proven using the compound angle formulae of trigonometry, and is thus reasonably accessible to students in senior calculus courses. See These proofs are best written as proofs by contradiction.

The original theorem is used in the proof of each converse theorem. We have seen this approach when Pythagoras’ theorem was used to prove the converse of Pythagoras’ theorem. The converse of the angle at the centre theorem The key theorem of the present module is, ‘An angle at the circumference of a circle is half the angle at the centre subtended by the same arc.’ The most straightforward converse of this is: Theorem Let OAB be isosceles with OA = OB, and let P be a point on the same side of the interval AB as O. Suppose that AB subtends an angle at P half the size of the angle AOB. Then the circle with centre O through A and B passes through P . Proof Let APB = a and AOB = 2a. Construct the circle through A and B. Suppose, by way of contradiction, that the circle does not pass through P, and let the circle cross AP, produced if necessary, at X. Join BX. Then (corresponding angles are equal); so the lines BX and PX coincide, and hence the points X and P coincide. Thus P lies on the circle, contradicting the assumption that the circle does not pass through P.

Testing whether four given points are concyclic One of the basic axioms of geometry is that a line can be drawn through any two distinct points A and B. When there are three distinct points A, B and C, we can ask whether the three points are collinear, or form a triangle. Similarly, any three non-collinear points A, B and C are concyclic. When there are four points A, B, C and D, no three collinear, we can ask whether these four points are concyclic, that is, do they lie on a circle. The remaining converse theorems all provide tests as to whether four given points are concyclic. In each case, we draw the unique circle through three of them and prove that the fourth point lies on this circle. We shall assume that the fourth point does not lie on the circle and produce a contradiction. The converse of the angles on the same arc theorem Theorem If an interval subtends equal angles at two points on the same side of the interval, then the two points and the endpoints of the interval are concyclic. Proof Let the interval AB subtend equal angles a at points P and Q on the same side of BC.

Construct the circle through A, B and P, and suppose, by way of contradiction, that the circle does not pass through Q. Let AQ, produced if necessary, meet the circle again at X. (corresponding angles are equal) Hence BQ and BX are the same line, so Q coincides with X, which is a contradiction. A test for a cyclic quadrilateral We proved earlier, as extension content, two tests for a cyclic quadrilateral: If the opposite angles of a cyclic quadrilateral are supplementary, then the quadrilateral is cyclic. If an exterior angle of a quadrilateral equals the opposite interior angle, then the quadrilateral is cyclic. The proof by contradiction of the first test is almost identical to the proof of the previous converse theorem. The second test is a simple corollary of the first test. The intersecting chord test We proved that the product of the intercepts of intersecting chords are equal. The converse of this gives yet another test for four points to be concyclic. The proof proceeds along exactly the same lines. Theorem Let AB and PQ be intervals intersecting at M, with AM × BM = PM × QM. Then the four endpoints A, B, P and Q are concyclic.

EXERCISE 28 Complete the proof of this theorem. We leave it the reader to formulate and prove the (true) converses to the remaining two theorems about secants from an external point, and a tangent and a secant from an external point. By similar methods, one can also prove the converse of the theorem that if a quadrilateral has an incircle, then the sums of its opposite sides are equal. AMO. EXERCISE 3 All radii of a circle are equal, so OA = OB = OP = OQ, so the quadrilateral APBQ is a rectangle because its diagonals are equal and bisect each other. Hence APB is a right angle. EXERCISE 4 At all times, the front of the building is the hypotenuse of aright-angled triangle whose third vertex is the photographer. Hence the circle with diameter, the front of the building, always pass through the photographer, and his possible positions are the points on the semicircle in front of the building. EXERCISE 5 At all times, the triangle formed by the plank, the wall and the floor is right-angled with hypotenuse of length metres.

By the converse theorem above, the midpoint of the plank is therefore always metres from the corner. Hence the midpoint traces out a quadrant of the circle with centre at the corner and radius metres. Case 2: It will be sufficient to prove the result when O lies on AP. Let b The first diagram below illustrates the situation with two indirect common tangents. The indirect tangents AT and BU intersect at M. Then AM = BM and TM = UM because tangents from an external point are equal, and adding the lengths gives AT = BU. c With direct common tangents, there are two cases to consider. Suppose first that the direct tangents AT and BU intersect at M when produced, as in the second diagram above. Then again AM = BM and TM = UM, and subtracting gives AT = BU. Now suppose that the direct common tangents AT and BU are parallel, as in the third diagram. The radii AO and BO are perpendicular to the parallel tangents, so they lie on the one line and form a diameter, and the radii TZ and UZ also form a diameter. Hence ABUT is a rectangle, so its opposite sides AT and BU are equal.

EXERCISE 18 This follows by a similar argument to parts a − c. e This follows because the diagonals of each rectangles are equal and bisect each other. Note that FQ is a diagonal of both rectangles. f Since N is the midpoint of the hypotenuse of the right-angled triangle FQV, it follows by the converse of the angle-in-a-semicircle theorem, proven earlier in this module, that N is equidistant from F, Q and V. Similarly, N is equidistant from E, P and U, and from G, R and W. Hence the circle passes through all nine points. g i The perpendicular bisector of UP is the locus of all points equidistant from U and P, and this includes the nine-point centre N. ii Complete the rectangle UPOT. Then NS || HTU, so NS bisects OT using opposite sides of rectangles, and so also bisects OH using a similarity theorem on triangles. Hence NS meets OH at Y . iii By similar arguments to parts i and ii, the perpendicular bisectors of UP, VQ and WR all pass through N, and all bisect OH.

Hence the three perpendicular bisectors are concurrent, and the midpoint of OH and the point N coincide at the intersection of these three lines. EXERCISE 28 Construct the circle through A, B and P, and suppose by way of contradiction that the circle does not pass through Q. Let PQ, produced if necessary, meet the circle again at X . Then XM = QM, and Q and X both lie on the same side of M. Hence X coincides with Q, which is a contradiction. The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Question: What is the straight line called which joins two points on the circumference of a circle but does not pass not through the centre?
Answer: CHORD