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 geometrical diagram showing a circle inside a triangle inside a larger circle.](http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/Conway_circle_theorem.svg/300px-Conway_circle_theorem.svg.png)
Proof
editLet I be the center of the incircle of triangle ABC, r its radius and Fa, Fb and Fc 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 IFa, IFb and IFc are perpendicular to a, b and c. Furthermore the following equalities for line segments hold. |AFc|=|AFb|, |BFc|=|Ba|, |CFa|=|Cb|. With that the six triangles IFcPa, IFcQb, IFaPb, IFaQc, IFbQa and IFbPc all have a side of length |AFc|+|BFc|+|CFa| 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 |IPa|=|IQa|=|IPb|=|IQb|=|IPc|=|IQc|. So the six points Pa, Qa, Pb, Qb, Pc and Qc have all the same distance from the triangle incenter I, that is they lie on a common circle with center I.
Additional properties
editThe radius of the Conway circle is
where and are the inradius and semiperimeter of the triangle.[3]
Generalisation
editConway'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.
See also
editReferences
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".
- 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.