Posts tagged 'carnot'

F.4 Carnot’s Theorem

In any triangle ABC, the sum of the signed perpendicular distances from the circumcenter O to the three sides BC, CA, AB equals the sum of the circumradius R and the inradius r: d_a + d_b + d_c = R + r.

F.38 Pedal Triangles

The pedal triangle A’B’C’ of a point P(tau_A : tau_B : tau_C) has area Delta(P) = ((R^2 - OP^2)/(4 R^2)) Delta, and its six side-projection segments satisfy Carnot’s alternating square identity AC’^2 + BA’^2 + CB’^2 = C’B^2 + A’C^2 + B’A^2.

F.45 Squares, Subdivided Lattices, and the Japanese Theorem

A square inscribed in a circle of radius R has side s = sqrt(2) R; inside a square of side s, intersecting cevians divide the interior in exact rational proportions, and by Carnot’s Theorem the sum of inradii in any triangulation of a cyclic polygon is invariant.