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

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.

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. 1 2 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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