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 geometrical diagram showing a circle inside a triangle inside a larger circle.
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

The radius of the Conway circle is

where and are the inradius and semiperimeter of the triangle.[3]

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]

See also

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


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