Conway circle theorem

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.^[1][2][3] The theorem and circle are named after mathematician John Horton Conway.

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

Proof

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

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

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

where ${\displaystyle r}$ and ${\displaystyle s}$ are the inradius and semiperimeter of the triangle.^[3]

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.^[4]

If 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