Carnot's theorem

Carnot's theorem gives a condition for when three points, each lying on one side of a triangle (or its extension), all lie on a single circle (are concyclic) with the triangle's vertices. It uses the distances from the triangle's vertices to these points.

Important properties

• If points $A'$, $B'$, $C'$ lie on sides $BC$, $CA$, $AB$ of triangle $ABC$ (or their extensions), then $A'$, $B'$, $C'$ are concyclic with $A$, $B$, $C$ if and only if a certain product of directed segment lengths equals $-1$.

• The theorem is useful for proving concyclicity in olympiad geometry problems.

• It generalizes the idea that four points are concyclic if and only if a certain power-of-point relation holds.