Posts tagged 'ceva'

F.7 Ceva and Menelaus

Three cevians AD, BE, CF of triangle ABC concur at a point P if and only if (AF/FB)(BD/DC)(CE/EA) = +1 (Ceva’s Theorem), whereas points D, E, F’ on the sidelines are collinear if and only if (AF’/F’B)(BD/DC)(CE/EA) = -1 (Menelaus’s Theorem), making (A, B; F, F’) a harmonic range.

F.47 The UVW Coordinate Triangle

Placing the altitude foot H_C at the origin (0, 0) with A = (-U, 0), B = (W, 0), and C = (0, V) parameterizes any triangle by three numbers (U, V, W) with a = sqrt(V^2 + W^2), b = sqrt(U^2 + V^2), c = U + W, and reduces the cevian traces D, E, F through P(p, q) to explicit 2x2 rational matrix formulas.