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Conway’s Circle Theorem: A Short Proof, Enabling

Generalization to Polygons

Eric Braude

MET Computer Science Department

Boston University

Boston, USA

orcid.org/0000-0002-1630-5509

Abstract— John Conway’s Circle Theorem is a gem of plane

geometry: the six points formed by continuing the sides of a

triangle beyond every vertex by the length of its opposite side,

are concyclic. The theorem has attracted several proofs, even

adorned Mathcamp T-shirts. We present a short proof that

views the extended sides as equal tangents of the incircle, a

perspective that enables generalization to polygons.

Keywords—geometry, Conway circle theorem, triangles

concurrency

I. INTRODUCTION

Several proofs of Conway’s Circle Theorem exist. A

recent proof by distinguished geometer Doris Schattschneider

[7] presents what she calls “a nicer, more convincing” proof

than her first—which had already been praised for its

"aesthetic appeal, … perhaps the one Euclid would have

worked out had he noticed Conway's theorem himself” [7].

Schattschneider’s second proof is based on elegant

constructions joining points of the hexagon. In May 2020,

Aperiodical published a proof without words by Colin

Beveridge [1]. In [7], Baker calls this "one of the most

beautiful Proofs Without Words I've ever seen". It constructs

line segments joining the six points to the incentre. A video

with words appeared in June 2020 [2]. Alex Ryba [8]

produced a short proof appealing to simple constructions and

overlapping isosceles triangles.

This paper is based on the following proof of Conway’s Circle

Theorem for triangle ABC with respective opposite side

lengths a, b, and c, as shown in Fig. 1.

a

Ab

c

b

c

B

C

a

b

ac

Fig. 1. Conway's Circle Theorem.

II. PROOF OF CONWAY’S CIRCLE THEOREM

As shown in Fig. 2, let I be the incenter of ABC, PCA the

point on the extension of CA at a distance a from A, and Ω

the circle with center I containing PCA. Let PCB, PAB, PAC, PBC,

and PBA be the intersections of Ω with the extensions of sides

CB, AB, AC, BC, and BA respectively. Chords PCAPAC,

PABPBA, and PBCPCB are rotations of each other around I, and

are thus equal. Since each chord pair is symmetric about the

diameter through its intersection, APBA = a, CPBC = c', and

BPAB = b'. The chord equality thus yields b' + a + c' = b' + c

+ a = c' + b + a. Thus, c' = c, b' = b, and Conway’s theorem

follows.

a

P

CA

A

Ω

b

c

b'

c'

I

zdenotes a given quantity

P

BA

P

BC

P

AC

P

AB

P

CB

B

C

a

Fig. 2. Proof of Conway's Circle Theorem.

Generalizations of Conway’s Circle Theorem have been

investigated by several authors; for example, Capitan [3],

who extended it to conics. The generalization in Theorem 1

below refers to the circles centered at the incenter and

containing the incircle. We will call these Conway Circles.

Theorem 1 reduces to Conway’s Circle Theorem when

xA

=

λA, xB

=

λB, and xC

=

λC. In this case, the chord lengths

equal the perimeter, an observation that will echo in

subsequent theorems.

III. THEOREM 1: CONWAY CIRCLES FOR TRIANGLES

Let T be a triangle with vertices A, B, and C, and

respective opposite side lengths λA, λB, and λC. Suppose that

the two sides passing through A (resp. B, C) are extended by

a distance xA (resp. xB, xC), as illustrated in Fig. 3. The six

endpoints of these extensions lie on a common circle

centered at the incenter of T if and only if xU – xV = λU – λV

for all vertices U and V of T.

λ

Α

x

A

x

B

x

C

A

x

A

x

B

x

C

C

B

λ

Β

λ

Χ

Fig. 3. Conway circles for triangles.

IV. PROOF OF THEOREM 1: CONWAY CIRCLES FOR TRIANGLES

Let I be T’s incenter, A' the point on the extension of

CA defined by xA, and Ω the circle centered at I that

contains A'. For each pair of vertices U and V, let yUV be

the segment length on the extension of UV defined by V

and

Ω

, as shown in Fig. 4.

xA

A' A

Ω

yBA

IλA

C

B

λB

λC

yBC

yAC

yAB

yCB

Fig. 4. Notation for proof of Theorem 1.

Since the three chords are tangential to the incircle, they

are rotations of each other about I, and thus of equal length.

Since each pair of these chords is symmetrical about the

diameter through its intersection, yBA = xA, yBC = yAC, and yAB

= yCB. We can thus label yBC and yAC as yC; also, yAB and yCB as

yB. Thus,

yB + λA + yC = xA + λB + yC

and so

yB = xA + λB – λA

Similarly,

yU = xV + λU – λV for all vertices U and V of T (1)

To prove sufficiency: given xA, xB, and xC satisfying xV -

xW = λV - λW for all vertices V and

Ω

of T, we have, using (1),

xB = xA + λB - λA = yB

Similarly, xC = yC , so

Ω

is a circle centered at I, passing

through the six points illustrated in Fig. 3.

Conversely, if there is a circle with incenter I intersecting

the six points specified by xA, xB, and xC, then it must be

Ω

because the latter is the unique circle centered at the incenter

and intersecting at A', which is defined by xA. Consequently,

xZ = yZ for Z = A, B, and C; and xU – xV = yU – yV = λU – λV for

all vertices V and Ω of T by (1).

The circle

Ω

in Theorem 1 reduces to the incircle when

(– xA) + (– (λB – λA + xA)) = λC, i.e., xA = ½ (λA – λB – λC).

For every xA < ½ (λA – λB – λC), the resulting circle

coincides with one for which xA > ½ (λA – λB – λC) and so we

can assume the latter. Since the results concern all circles at

the incenter at least as large as the incircle, we will refer to

them as “Conway circles.”

Theorem 1 provides corollaries by selecting interesting

values of xV. For example, Corollary 1 follows by taking xA =

0, and is illustrated in Fig. 5 for λB > λA and λC < λA.

V. COROLLARY 1: CONWAY CIRCLES AT TRIANGLE

VERTICES

Given a triangle with vertices A, B, and C, and respective

opposite side lengths λA, λB, and λC, if sides are extended by

λB – λA at B and by λC – λA at C, then the resulting five points,

including A, are concyclic, with center at the incenter.

λA

λB

λC

A

λC–λA

λB–λA

B

C

Fig. 5. Conway circles at a vertex.

A tangential polygon is one for which a circle exists

whose sides are tangential to it. Every convex polygon P

corresponds to a set of tangential polygons with sides that are

pairwise parallel to P. (On any circle C, construct successive

tangents to C, each parallel to successive sides of P.) We will

use the chord symmetry argument described above for

tangential polygons with an odd number of sides (“odd

polygons”). We then use this result to generalize for even

tangential polygons. Theorem 2 generalizes Theorem 1. Its

proof proceeds like that for Theorem 1, progressing around

the polygon once for odd indices and then, because n is odd,

continuing in the same manner for even indices. This process

uses a modified mod function

µ

().

VI. THEOREM 2: CONWAY CIRCLES FOR ODD TANGENTIAL

POLYGONS

Let P be a tangential polygon with an odd number n of

vertices V1, V2, … , Vn, define li as ViVi+1 for i = 1, 2, …, n-1

and λn as VnV1, illustrated in Fig. 6. Suppose that each pair of

sides ending at Vi are extended by length xi. The 2n endpoints

of these extensions lie on a common circle centered at the

incenter of P if and only if

xi – x

µ

(i+2, n) = λ

µ

(i+1, n) – λi for 0 < i ≤ n (2)

—where

µ

(z, m) is defined as z mod m for z ≠ m, and m

otherwise.

V

2

λ

n

x

n

x

n

x

1

x

1

V

1

zdenotes a

given quantity

x

i

x

i+1

λ

i

x

i

V

i+1

λ

1

λ

n-1

x

i+1

λ

i+1

V

i

Fig. 6. Conway circles for odd tangential polygons.

VII. PROOF OF THEOREM 2: CONWAY CIRCLES FOR ODD

TANGENTIAL POLYGONS

As illustrated in Fig. 7, let I be the incenter of the given

polygon, V1' the point on the extension of λ1 from V1 defined

by x1, and Ω the circle, centered at I, containing V1'. For 1 ≤ i

≤ n, let yi and yi' be the segment lengths on the side extensions

through Vi defined by their intersections with Ω.

V

i+1

λ

n

y

n

y

n

'

x

1

y

1

V

1

y

i

'

y

i+1

'

y

i+1

λ

i

y

i

V

i+1

V

2

Ω

V

1

'

I

λ

1

λ

i+1

V

i

λ

n-1

λ

i-1

Fig. 7. For proof of Conway Circles for odd tangential polygons

As in the proof of Theorem 1, y1 = x1, yi = yi' for all 2 ≤ i

≤ n, and the n chords are equal. Thus, for 1 ≤ i ≤ n – 2,

yi + λi + yi+1 = yi+1 + λi+1 + yi+2

and so yi – yi+2 = λi+1 – λi

For 1 ≤ i < n – 2 and i = n, this establishes

yi – y

µ

(i+2, n) = λ

µ

(i+1, n) – λi (3)

Equation (3) is also satisfied for i = n – 1 and i = n – 2,

since it reduces to, respectively,

yn-1– y

µ

(n+1, n) = λ

µ

(n, n) – λn-1

and yn-2 – y

µ

(n, n) = λ

µ

(n-1, n) – λn-2

i.e., yn-1– y1 = λn – λn-1

and yn-2 – yn = λn-1 – λn-2

But these follow from the chord length equalities

yn-1 + λn-1 + yn = yn + λn + y1 and yn-2 + λn-2 + yn-1 = yn-1 +

λn-1 + yn resp.

To prove sufficiency in Theorem 2, assume that there is a

circle V, centered at the incenter, that intersects the sides

extended from Vi at distances xi for 1 ≤ i ≤ n. Thus, V=W

because W is defined by the incenter and the distance x1. The

relationships (3) follow as above for Theorem 1, which

concludes the sufficiency.

To prove the necessity, assume equation (2) for some x1, x2, …,

xn,. We thus have x1 – x

µ

(3, n) = λ

µ

(2, n) – λi

Constructing W from x1 as above, the consequence x1 = y1,

and equation (3) together imply

λ

µ

(2, n) – λi = x1 – y

µ

(3, n)

so x3 = y3. Similarly, x5 = y5, x7 = y7, … , and xn = yn.

For i = n, equation (2) becomes

xn – x

µ

(n+2, n) = λ

µ

(n+1, n) – λn

i.e., since xn = yn, we have yn – x2 = λ1 – λn .

But the equality of the chords in Fig. 7 implies

yn + λn + y1 = y1 + λ1 + y2

so we have yn – x2 = yn – y2. Thus, x2 = y2, and the

corresponding equalities continue for the even indices.

Corollary 2 below is easily recognized as a direct

generalization of Conway’s Circle Theorem for triangles, the

chord length equaling the perimeter of the polygon. It follows

from Theorem 2 by verifying equation (2).

VIII. COROLLARY 2: CONWAY’S CIRCLE FOR ODD POLYGONS

Let P be a tangential polygon with vertices V1, V2, … ,

Vn, n odd, and λi denoting ViVi+1 for i = 1, 2, 3, …, n, as

shown in Fig. 8. Suppose that each pair of sides ending at Vi

are extended by length Σ{

λ

i: (1≤i<k

∧

i odd) ∨ (k<i≤ n ∧ i

even)} for odd k, and Σ{ λi: (1≤i<k ∧ i even) ∨ (k<i≤n ∧ i

odd)} for even k. Then the 2n endpoints of these extensions

lie on a common circle centered at the incenter of P.

λk

for odd k:

Σ{λi: (1≤i<k ∧iodd) ∨(k<i≤n ∧ieven)}

Σ{λi: (1≤i<k ∧ieven) ∨(k<i≤n∧iodd)}

Vk

Vk+1

Fig. 8. Conway’s circle for odd tangential polygons.

Corollary 3 is the application of Corollary 2 to pentagons,

in which the generalization of Conway’s Circle Theorem is

quite graphic. It follows by taking (λ1, λ2, λ3, λ4, λ5) = (a, b,

c, d, e).

IX. COROLLARY 3: CONWAY’S CIRCLE FOR TANGENTIAL

PENTAGONS

Let ABCDE be a tangential pentagon with opposite side

lengths a, b, c, d, and e respectively. If the sides are

extended at A, B, C, D, and E by b + e, a + c, b + d, e + c,

and a + d respectively, then the 10 resulting points are

concyclic, with center at the pentagon’s incenter. Fig. 9

illustrates this.

a

b

c

e

d

a + c

c + e

b + d

a + d

b + e

C

D

E

A

B

Fig. 9. Conway’s Circles for tangential pentagons.

We will produce Conway circles next for even tangential

polygons. Given an even sequence of lengths that form a

tangential polygon, there are infinitely many such tangential

polygons, as shown in [6]. The Conway circle formulae are

slightly more complicated as a result. We will confine our

result to necessary conditions, which continue to generalize

the Conway Circle Theorem for triangles.

X. THEOREM 3: CONWAY CIRCLES FOR EVEN TANGENTIAL

POLYGONS

Given an even-sided tangential polygon V1, V2, …, Vn,

with h1 defined as V1V2, h2 as V2V3, …, and hm as VmV1, in

which the incircle is incident on VmV1 at a distance h0 from

V1, extend each side VmVm+1, with odd m, by quantity (4)

below at Vm and by quantity (5) at Vm+1, then the 2n points so

formed are concyclic, with center at the polygon’s incenter.

This is illustrated in Fig. 10.

Σ{hi: (1≤i<k ∧ i odd) ∨ (k<i≤m ∧ i even)} – h0 (4)

Σ{hi: (1≤i<k ∧ i even) ∨ (k<i≤m ∧ i odd )} + h0 (5)

for odd k

hk

h1

h0

hm

Σ{hi: (1≤i<k ∧iodd) ∨(k<i≤m∧ieven)} –h0

Σ{hi: (1≤i<k ∧ieven) ∨(k<i≤m∧iodd )} + h0

Vk

V1

Fig. 10. Conway’s circle for even tangential polygons.

XI. PROOF OF THEOREM 3: CONWAY CIRCLES FOR EVEN

TANGENTIAL POLYGONS

As shown in Fig. 11, we introduce a distance h0' from V1

on V1V2 a little further than h0 (h0' will converge to h0),

defining the point V2', then continuing the existing tangential

polygon from V2' instead of V2. This replaces the segments

V1V2 and V2V3 with segments V1V2' , V2'V2'', and V2''V3. These

have lengths h0', h1', and h2' respectively, say.

V

1

Angle ε

h

0

h

2

'

h

m

h

1

h

0

'

h

1

'

h

k

(h

1

' + h

3

+ h

5

+ … + h

k-2

)

+ (h

k+1

+ h

k+3

+ … + h

m

)

h

2

(h

2

' + h

4

+ h

6

+ … + h

k-1

) +

(h

k+2

+ h

k+4

+ … + h

m-1

+ h

0

')

V

2

V

2

'

V

2

'

'

zdenotes a given quantity

for odd k

Fig. 11. Proving Conway’s circle for even tangential polygons.

The resulting polygon has an odd number of sides, so we

can apply Corollary 2 with h0' replacing λn of Theorem 2, h1'

replacing λ1, h2' replacing λ2, and hi replacing λi for the

remaining sides. The expressions shown in Figure 11 result.

As angle ε→0, we have h0'

→

h0, h1'

→

h1 - h0, h2'

→

h2, and the

following limits hold for the two expressions displayed in

Fig. 11, proving Theorem 3:

(h1' + h3 + h5 + … + hk-2) + (hk+1 + hk+3 + … + hm)

→ Σ(hi: (1≤i<k ∧ i odd) ∨ (k<i≤m ∧ i even)} – h0

and

(h2' + h4 + h6 + … + hk-1) + (hk+2 + hk+4 + … + hm-1 + h0')

→ Σ{hi: (1≤i<k ∧ i even) ∨ (k<i≤m ∧ i odd)} + h0

Corollary 4 follows by taking n = 4, and h1, h2, h3, and h4

= a, b, c, and d respectively in Theorem 3.

XII. COROLLARY 4: CONWAY CIRCLE FOR TANGENTIAL

QUADRILATERALS

For any tangential quadrilateral with side lengths a, b, and

c, and the tangency on the remaining side—with length d—

at a distance d0 from a vertex, extend the sides between a and

b, b and c, c and d, and d and a, by c + d – d0 , a + d0 , b + d

– d0 , b + d – d0 , and b + d0 respectively. Then the resulting

eight points, as illustrated in Fig. 12, are concyclic.

a

b + d

0

b

c

a + d

0

b + d – d

0

d

c + d – d

0

d

0

Fig. 12. Conway circles for a tangential quadrilateral.

In summary, when we view the side extensions in Conway’s

pretty circle theorem as equal chords tangential to the

incircle, a clear perspective emerges, generalizable to

tangential polygons

Acknowledgment

The author is grateful to Matt Baker for his

encouragement and for his elegant reformulation of Theorem

1.

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