Conway circle theorem

The Conway circle theorem states that the six points where a triangle’s incircle and three excircles touch its side lines lie on a single circle.

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Triangle: A polygon with three sides, three vertices, and three interior angles. The theorem’s six contact points are defined using a triangle’s side lines.
Tangent line: A line that meets a circle at exactly one point and is perpendicular to the radius there. Each contact point is defined by a side line tangent to an incircle or excircle.
John Horton Conway: An English mathematician known for contributions to group theory, combinatorics, geometry, and recreational mathematics. The theorem bears Conway’s name, linking its circle configuration to his geometric work.
Equilateral triangle: A triangle whose three sides and three angles are equal. Its symmetry makes the six tangency points form a regular hexagonal arrangement on the circle.
Incircle: The circle inside a triangle that is tangent to all three side segments. Its three tangency points supply half of the six points in the theorem.
Semiperimeter: Half the perimeter of a polygon; for a triangle with sides a, b, and c, it is (a+b+c)/2. The theorem’s contact distances can be expressed using the triangle’s semiperimeter.
Triangle geometry: The study of properties and constructions associated with triangles. The theorem belongs to the broader tradition of discovering invariant configurations in triangles.
Right triangle: A triangle with one interior angle measuring 90 degrees. Its side lengths provide a simple setting for calculating the incircle and excircles explicitly.
Excircle: A circle tangent to one side of a triangle and the extensions of its other two sides. The three excircles supply the other three tangency points.
Inradius: The radius of a triangle’s incircle. The incircle’s contact geometry depends on its radius and the triangle’s area.