GeoProof — Carnot's Theorem explorer

Carnot's sum of signed distances

Carnot's Theorem

Drop a perpendicular from the circumcenter O onto each side of a triangle. Carnot's theorem says the three (signed) lengths always add up to the circumradius plus the inradius:

d_a + d_b + d_c = R + r

Each foot lands exactly on the midpoint of its side — that's just because O is equidistant from the two endpoints of every chord of the circumcircle.

The signs

A distance is counted as negative when O lies on the opposite side of that edge from the triangle. This happens for the side opposite an obtuse angle, since an obtuse triangle's circumcenter falls outside it. With the signs included, the identity holds for every triangle.

Why it works

The signed distance from O to the side opposite vertex A is exactly R·cos(A). So the sum is R(cos A + cos B + cos C). A standard identity says cos A + cos B + cos C = 1 + r/R, and multiplying by R gives R + r.

Right triangles

When one angle is 90°, its cosine is 0, so the perpendicular to the hypotenuse has length 0 — the circumcenter sits exactly on the hypotenuse's midpoint. The other two distances then add to R + r on their own.

A bit of history

It's named for the French mathematician and statesman Lazare Carnot (1753–1823), who developed signed-length methods in his work on the geometry of position.