The Improving Mathematics Education in Schools (TIMES) Project
The Improving Mathematics Education in Schools (TIMES) Project
Measurement and Geometry : Module 26Years : 9-10
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.
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.
Symmetries of a circle
Circles have an abundance of symmetries:
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.
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:
Let AB be a chord of a circle with centre O. The following three lines coincide:
Trigonometry and chords
Constructions with radii and chords give plenty of opportunity for using trigonometry.
In the diagram to the right, the interval AM subtends an angle θ
at the centre O of a circle of radius 1. Show that:
a AM = sin θ b 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.
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.
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.
An angle in a semicircle is a right angle.
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 = B.
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)
|2α + 2β||= 180°|
|α+ β||= 90°,|
|so||APB||= 90°, as required.|
This famous theorem is traditionally ascribed to the Greek mathematician Thales, the first known Greek mathematician.
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.
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:
The circle whose diameter is the hypotenuse of a right-angled
triangle passes through all three vertices of the triangle.
Let 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.
A photographer is photographing the ornamental front
of a building. He wants the two ends of the front to
subtend a right angle at his camera. Describe the set
of all positions where he can stand.
A plank of length metres is initially resting flush against a wall, but it slips outwards, with its top sliding down the wall and its foot moving at right angles to the wall. What path does the midpoint of the plank trace out?
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.
An angle at the circumference of a circle is half the angle at the centre subtended by the same arc.
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 = 2APB. 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||OAP||= α and OBP = β.|
|Then||APO||= α||(base angles of isosceles OAP)|
|and||BPO||= β||(base angles of isosceles OBP).|
|Hence||AOQ||= 2α||(exterior angle of OAP)|
|and||BOQ||= 2β||(exterior angle of OBP),|
|AOB||= 2α + 2β = 2(α + β) = 2 × APB.|
Complete the proof in the other two cases.
A punter stands on the edge of a circular racing track. With his binoculars he is following a horse that is galloping around the track at one revolution a minute. Explain why the punter’s binoculars are rotating at a constant rate of half a revolution per minute.
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 AQB
are subtended by the same (minor) arc AB. Each angle is half
the angle AOB at the centre subtended by the same arc,
so APB = AQB.
This corollary of the previous theorem is a particularly significant
result about angles in circles:
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.
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.
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.
The altitudes of a triangle are concurrent.
In the diagram to the right, the altitudes AP and
BQ meet at H. The interval CH is produced to meet AB , produced if necessary, at R. We need to prove that CR ⊥ AB.
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.
The opposite angles of a cyclic quadrilateral are supplementary.
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||α+ γ||= 360°||(angles in a revolution at O)|
|Also||A||= α||(angles on the same arc BCD)|
|and||C||= γ||(angles on the same arc BAD)|
|so||A + C||= α + γ= 180°|
|Hence also ABC + ADC = 180° (angle sum of quadrilateral ABCD)|
Here is an alternative proof using the fact that two angles in the same segment are equal.
Join the diagonals AC and BD of the cyclic quadrilateral ABCD. Let α, β, γ and δ be the angles shown.
Exterior angles of a cyclic quadrilateral
An exterior angle of a cyclic quadrilateral is supplementary to the adjacent interior angle, so is equal to the opposite interior angle. This gives us the corollary to the cyclic quadrilateral theorem:
An exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
In the diagram to the right, BC is produced to P to form the exterior angle PCD. This exterior angle and A are both supplementary to BCD, so they are equal.
Show that AP || CR in the diagram to the right.
If a cyclic trapezium is not a rectangle, show that the other two sides are not parallel, but have equal length.
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.
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 A and BXD are supplementary because ABXD is a cyclic quadrilateral,
so the angles DXB and 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.
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.
In the diagram to the right, the two adjacent acute angles of
the trapezium are equal.
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,
In any triangle ABC, = = = 2R, where R is the radius of the circumcircle.
Let O be the centre of the circumcircle through A, B and C, and let A = α.
It is sufficient to prove that is the diameter of the circumcircle.
There are three cases, as shown below.
|Case 1: A and O lie onthe same side of BC.||Case 2: A and O lie on opposite sides of BC.||Case 3: O lies on BC|
In cases 1 and 2, construct the diameter BOP, and join PC .
a Complete the following steps of the proof in Case 1.
i Explain why BCP = 90° and P =α
iiComplete the proof using BCP.
b Complete the following steps of the proof in Case 2.
i Explain why BCP = 90° and sin P = sinα
iiComplete the proof using BCP.
c 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 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.
Let T be a point on a circle with centre O.
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 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
Tangents to a circle from an external point have equal length.
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 .
A tangent PT to a circle of radius 1 touches the circle
at T and subtends an angle θ at the centre O. Show that
PT = tan θ and PO = sec θ
where the trigonometry function secant is defined
by sec θ = .
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.
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.
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.
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.
We have to prove IC bisects BCD
Given ABC , construct the angle bisectors of A and B, and let I be their point of intersection.
Join IC, and let α = BAI = CAI and β = ABI = CBI.
Drop perpendiculars IL, IM and IN to BC, CA and AB respectively.
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 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.
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.
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 P = α
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
|Case 1||Case 2||Case 3|
Case 1 is easily settled:
α = 90° (angle in a semicircle) and ABU = 90° (radius and tangent)
For the other two cases, construct the diameter BOD, and join DA.
|Case 2:||ADB||= α||(angles on the same arc AB)||
|and||BAD||= 90°||(angle in a semicircle)|
|so||DBA||= 90° − α||(angle sum of ABD)|
|But||DBU||= 90°||(radius and tangent)|
|so||ABU||= α||(adjacent angles at B)|
|Case 3:||ADB = 180° − α||(cyclic quadrilateral APBD)||
|and||BAD = 90°||(angle in a semicircle)|
|so||ABD = α− 90°||(angle sum of ABD)|
|But||DBU = 90°||(radius and tangent)|
|so||ABU = α||(adjacent angles at B)|
Find an alternative proof in cases 2 and 3 by constructing the radii AO and BO and using angles at the centre.
The result in the following exercise is surprising. One would not expect parallel lines to emerge so easily in a diagram with two touching circles.
Prove that AB || PQ in the diagram to the right,
where STU is a common tangent to both circles.
AQ and BP are straight lines.
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.
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.
When two chords intersect within a circle, the products of the intercepts are equal.
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 APM is similar to QBM (AA):
Hence = (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.
Use the sine rule in the diagram in the above proof to prove that = .
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:
The product of the intercepts on two secants from an external point are equal.
Let ABM and PQM be two secants from an external point M, as shown.
Join AP and BQ.
Complete the proof.
If instead we had joined the intervals AQ and BP, what corresponding changes should be made
to the proof?
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.
The product of the intercepts on a secant from an external point equals the square of the tangent from that point.
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.
The concurrence of three common chords
Proving concurrence usually involves logic that is a little
more sophisticated than required in most proofs. The
following theorem is a difficult application of the
intersecting chord theorem.
Show that the three common chords AB, PQ and ST to the three circles in the diagram
above are concurrent.
Circle geometry is often used as part of the solution to problems in trigonometry and calculus.
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.
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||= 2|
|2x2 − 4x + 4||= 2|
|x2 − 2x + 1||= 0|
|(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|
|x2 − 2x + 1 = 0|
Hence P(1, 1) is the only point of intersection.
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.
Sketch the set of complex numbers such that the ratio is an imaginary number, that is a real multiple of i.
If z = 1, then = 0, which is imaginary.
If z = −1, then 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.
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.
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.
In the early 19th century, Poncelet, Feuerbach and others showed that in any triangle,
the following nine points are cyclic:
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.
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.
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.
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 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:
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 .
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||AXB||= AOB||(angles at centre and circumference on the same arc AB)|
|so||BX || BP||(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
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.
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.
|Then||AXB = α||(angles on the same arc AB),|
|so||BX || BQ||(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:
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.
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.
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.
Use simple trigonometry in AMO.
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.
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.
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.
|Then||BPO||= β||(base angles of isosceles OBP),|
|so||AOB||= 2β = 2 × APB||(exterior angle of OBP).|
Case 3: Join PO and produce PO to Q.
|and||OBP||= β||(base angles of isosceles OBP),|
|Then||APO||= α||(base angle of issosceles AOP)|
|OPB||= β||(base angle of OBP)|
|Hence||APB||= β − α|
|Also||AOQ||= 2α||(exterior angle of OAP)|
|and||BOQ||= 2β||(exterior angle of OBP)|
|Hence||AOB||= 2β − 2α = 2(β − α) = 2 × APB|
Join the common chord BQ, and produce ABC to X, and let a = A.
|Then||BQR = α||(opposite exterior angle of cyclic quadrilateral)|
|and||RCX = α||(opposite exterior angle of cyclic quadrilateral),|
|so||AP || CR||(corresponding angles are equal)|
In this proof, we construct two isosceles triangles.
Let ABCD be a cyclic trapezium with AB || DC. Since ABCD is not a rectangle, and its angles add to 360°, one of its angles is acute.
|Let||A = α be acute, and produce BC to P.|
|Then||DCP = α||(opposite exterior angle of cyclic quadrilateral)|
|so||B = α||(corresponding angles, AB || DC).|
|Hence||B is also acute, so AD and BC produced meet at a point M,|
|and||CDM = α||(corresponding angles, AB || DC).|
Hence DMC and AMB are both isosceles, with DM = CM and AM = BM , so AD = BC.
The adjacent interior angle is supplementary to the exterior angle, and therefore equal to the opposite interior angle. Hence the quadrilateral is cyclic by the theorem.
|D =||180° − α||(co-interior angles, AB || DC )|
|so||D + B =||180°|
Hence the quadrilateral ABCD is cyclic (opposite angles are supplementary).
|a||In Case 1,||BCP||= 90°||(angle in a semicircle)|
|and||P||= α||(angles on the same arc BC)|
|Hence||= sin α||(simple trigonometry in BCP)|
|b||In Case 2,||BCP||= 90°||(angle in a semicircle)|
|and||P||= 180° − α||(opposite angles of cyclic quadrilateral BACP)|
|so||sin P||= sin (180° − α ) = sin α|
|Hence||= sin α||(simple trigonometry in BCP)|
|c||In Case 3,||α||= 90°||(angle in a semicircle),|
|so||sin α||= 1.|
|The diameter of the circumcircle is a,|
|so||= = 2R.|
Let tangents from an external point P touch the circle at T and U.
Join the radii OT and OU and the interval OP.
Since OP is common, the radii are equal, and the radii are perpendicular to the tangents,
|so||PT||= PU||(matching sides of congruent triangles).|
The triangle OPT is right-angled at T. The rest is simple trigonometry.
Tangents from an external point are equal, so we can label the lengths in the figure as shown. Then both sums of opposite sides of the quadrilateral are a + b + c + d.
|In Case 2,||AOB||=||2α||(angles at centre and circumference on same arc AB)|
|so||OBA||=||90° −α||(angle sum of isosceles OAB)|
|But||OBU||=||90°||(radius and tangent)|
|so||ABU||=||α||(adjacent angles at B).|
|In Case 3,||reflex AOB||=||2α||(angles at centre and circumference on same arc AB)|
|so||non-reflex AOB||=||360° − 2α||(angles in a revolution)|
|OBA||=||α − 90°||(angle sum of isosceles OAB)|
|OBU||=||90°||(radius and tangent)|
|ABU||=||α||(adjacent angles at B).|
|Then||BTU||= α||(alternate segment theorem)|
|so||STP||= α||(vertically opposite angles)|
|so||Q||= α||(alternate segment theorem)|
|Hence AB || PQ||(alternate angles are equal).|
We have already proven that A = Q and that P = B.
The sine rule = in a triangle XYZ can also be written in the form
Using this form of the sine rule in the two triangles,
= = = .
Proof: In the triangles APM and QBM :
|PAM = BQM||(exterior angle of cyclic quadrilateral ABQP)|
|AMP = BMQ||(common angle)|
|so||APM is similar to QBM (AA).|
|Hence||= .||(matching sides of similar triangles)|
|so||AM × BM = PM × QM.|
Proof: In the triangles AQM and PBM :
|A = P||(angles on the same arc BQ)|
|M is common to both triangles|
|so||AQM is similar to PBM|
|Hence||=||(matching sides of similar triangles)|
|so||AM × BM = PM × QM|
Proof: In the triangles ATM and TBM:
|TAM = BTM||(alternate segment theorem)|
|AMT = TMB||(common)|
|so||ATM is similar to TBM (AA)|
|Hence||=||(matching sides of similar triangles)|
|AM × BM = TM2|
Let the chords AB and PQ meet at M. Join SM and produce it to meet the circles at X and Y. We need to prove that the points X and Y coincide.
Using the intersecting chord theorem in each circle in turn,
|SM × MX||= AM × MB||(from Circle 1)||
|= PM × MQ||(from Circle 2)|
|= SM × MY||(from Circle 3).|
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||AM × BM = PM × XM||(intersecting chords)|
|so||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.