Posts in category 'Cyclic Polygons & Four-Point Invariants'

W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation

Four non-collinear points A_1, A_2, A_3, A_4 on a circle c satisfy equal opposite spreads s(A_1 A_2, A_1 A_4) = s(A_3 A_2, A_3 A_4), Ptolemy’s Triple-Quad Law A(Q_12*Q_34, Q_23*Q_14, Q_13*Q_24) = 0 among product quadrances, and the cyclic quadrea / circumquadrance formulas, while four collinear points satisfy Q(Q_12, Q_23, Q_34, Q_14) = 0.