Carnot's theorem

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Let ABC be an arbitrary triangle. Then the sum of the signed distances from the circumcenter O to the sides of triangle ABC is

OO_A + OO_B + OO_C = R + r,

where r is the inradius and R is the circumradius. Here the sign of the distances is taken positive if and only if the line segment OO_i lies completely inside the triangle.

[edit] External links

• Carnot's Theorem at cut-the-knot
• Yet another Carnot's Theorem with multiple applications at cut-the-knot