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.

Proof edit

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_{a}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|=|B_a|, |CF_a|=|C_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 from the triangle incenter I, that is they lie on a common circle with center I.

Additional properties edit

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}{abc}}}

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

Generalisation edit

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 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.

References edit

^ "John Horton Conway". www.cardcolm.org. Archived from the original on 20 May 2020. Retrieved 29 May 2020.
^ Weisstein, Eric W. "Conway Circle". MathWorld. Retrieved 29 May 2020.
^ ^a ^b Francisco Javier García Capitán (2013). "A Generalization of the Conway Circle" (PDF). Forum Geometricorum. 13: 191–195.
^ Michael de Villiers (2023). "Conway's Circle Theorem as a Special Case of a More General Side Divider Theorem". Learning and Teaching Mathematics (34): 37–42.

External links edit

Kimberling, Clark. "Encyclopedia of Triangle Centers".
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.

[1] "John Horton Conway". www.cardcolm.org. Archived from the original on 20 May 2020. Retrieved 29 May 2020.

[2] Weisstein, Eric W. "Conway Circle". MathWorld. Retrieved 29 May 2020.

[FJGC-3] Francisco Javier García Capitán (2013). "A Generalization of the Conway Circle" (PDF). Forum Geometricorum. 13: 191–195.

[MdeV-4] Michael de Villiers (2023). "Conway's Circle Theorem as a Special Case of a More General Side Divider Theorem". Learning and Teaching Mathematics (34): 37–42.

[1]

[2]

[3]

[4]