Conway circle theorem

A triangle's Conway circle with its six concentric points (solid black), the triangle's incircle (dashed gray), and the centre of both circles (white); solid and dashed line segments of the same colour are equal in length

In plane geometry, the Conway circle theorem states that when the sides meeting at each vertex of a triangle are extended by the length of the opposite side, the six endpoints of the three resulting line segments lie on a circle whose centre is the incentre of the triangle. The circle on which these six points lie is called the Conway circle of the triangle. The theorem and circle are named after mathematician John Horton Conway.

Proof

segments of equal color are of equal length ${\begin{aligned}\triangle IF_{c}P_{a}&\cong \triangle IF_{c}Q_{b}\cong \triangle IF_{a}P_{b}\\&\cong \triangle IF_{a}Q_{c}\cong \triangle IF_{b}P_{c}\\&\cong \triangle IF_{b}Q_{a}\\\Rightarrow \,|IP_{a}|&=|IQ_{a}|=|IP_{b}|=|IQ_{b}|\\&=|IP_{c}|=|IQ_{c}|\end{aligned}}$

Let I be the center of the incircle of triangle ABC, r its radius and F_a, F_b and F_c the three points where the incircle touches the triangle sides a, b and c. Since the (extended) triangle sides are tangents of the incircle it follows that IF_a, IF_b and IF_c are perpendicular to a, b and c. Furthermore the following equalities for line segments hold. |AF_c|=|AF_b|, |BF_c|=|BF_a|, |CF_a|=|CF_b|. With that the six triangles IF_cP_a, IF_cQ_b, IF_aP_b, IF_aQ_c, IF_bQ_a and IF_bP_c all have a side of length |AF_c|+|BF_c|+|CF_a| and a side of length r with a right angle between them. This means that due SAS congruence theorem for triangles all six triangles are congruent, which yields |IP_a|=|IQ_a|=|IP_b|=|IQ_b|=|IP_c|=|IQ_c|. So the six points P_a, Q_a, P_b, Q_b, P_c and Q_c have all the same distance from the triangle incenter I, that is they lie on a common circle with center I.

Additional properties

The radius of the Conway circle is

${\sqrt {r^{2}+s^{2}}}={\sqrt {\frac {a^{2}b+ab^{2}+b^{2}c+bc^{2}+a^{2}c+ac^{2}+abc}{a+b+c}}}$

where $r$ and $s$ are the inradius and semiperimeter of the triangle.

Generalisation

Conway's circle theorem as a special case of the generalisation, called "side divider theorem" (Villiers) or "windscreen wiper theorem" (Polster))

Conway's circle is a special case of a more general circle for a triangle that can be obtained as follows: Given any △ABC with an arbitrary point P on line AB. Construct BQ = BP, CR = CQ, AS = AR, BT = BS, CU = CT. Then AU = AP, and PQRSTU is cyclic.

If you you place P on the extended triangle side AB such that BP=b and BP being completely outside the triangle then the above constructions yield Conway's circle theorem.

See also

• List of things named after John Horton Conway

External links

• Kimberling, Clark. "Encyclopedia of Triangle Centers".
• Colin Beveridge: Conway’s Circle, a proof without words. The Aperiodical, 07 May 2020
• Colin Beveridge, Elizabeth A. Williams: Conway’s Circle Theorem: a proof, this time with words. The Aperiodical, 11 June 2020 (Video, 9:12 min.)
• De Villiers, Michael. "Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem". dynamicmathematicslearning.com.
• Polster, Burkard (6 April 2024). "Conway's Iris and the Windscreen Wiper Theorem". Mathologer. YouTube.