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Conways Circle Theorem: A Short Proof Enabling Generalization to Polygons

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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. We present a short proof that views the extended sides as equal tangents of the incircle, a perspective that enables generalization to polygons.
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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
AbstractJohn 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.
Keywordsgeometry, 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
"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 CONWAYS 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
the circle with center I containing PCA. Let PCB, PAB, PAC, PBC,
and PBA be the intersections of 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
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 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
, as shown in Fig. 4.
xA
A' A
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 + λBλA
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
of T, we have, using (1),
xB = xA + λB - λA = yB
Similarly, xC = yC , so
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
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 xUxV = yUyV = λUλV for
all vertices V and of T by (1).
The circle
in Theorem 1 reduces to the incircle when
(– xA) + (– (λBλA + xA)) = λC, i.e., xA = ½ (λAλBλC).
For every xA < ½ (λAλB λC), the resulting circle
coincides with one for which xA > ½ (λAλBλC) 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
λCλA
λBλA
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
xix
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 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 .
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 n2,
yi + λi + yi+1 = yi+1 + λi+1 + yi+2
and so yi yi+2 = λi+1 λi
For 1 i < n2 and i = n, this establishes
yiy
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-1y
µ
(n+1, n) = λ
µ
(n, n) λn-1
and yn-2y
µ
(n, n) = λ
µ
(n-1, n) λn-2
i.e., yn-1y1 = λn λn-1
and yn-2yn = 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 x1x
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 = x1y
µ
(3, n)
so x3 = y3. Similarly, x5 = y5, x7 = y7, … , and xn = yn.
For i = n, equation (2) becomes
xnx
µ
(n+2, n) = λ
µ
(n+1, n) λn
i.e., since xn = yn, we have ynx2 = 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: CONWAYS 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: (1i<k
i odd) (k<i n i
even)} for odd k, and Σ{ λi: (1i<k i even) (k<in 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: (1i<k iodd) (k<in ieven)}
Σ{λi: (1i<k ieven) (k<iniodd)}
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: CONWAYS 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: (1i<k i odd) (k<im i even)} h0 (4)
Σ{hi: (1i<k i even) (k<im i odd )} + h0 (5)
for odd k
hk
h1
h0
hm
Σ{hi: (1i<k iodd) (k<imieven)} h0
Σ{hi: (1i<k ieven) (k<imiodd )} + 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: (1i<k i odd) (k<im i even)} h0
and
(h2' + h4 + h6 + … + hk-1) + (hk+2 + hk+4 + … + hm-1 + h0')
Σ{hi: (1i<k i even) (k<im 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
For any tangential quadrilateral with side lengths a, b, and
c, and the tangency on the remaining sidewith 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 + dd0 , a + d0 , b + d
d0 , b + dd0 , 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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Beveridge, C. Conway's Circle Theorem: a proof, this time with words, retrieved December 8, 2020 from https://aperiodical.com/2020/06/conways-circle-theorem-a-proof-thistime-with-words