Posts tagged 'projective'

F.2 Bilinear Transformations

The canonical bilinear transformation B(x) = (1 + x)/(1 - x) and its inverse B^{-1}(y) = (y - 1)/(y + 1) link the internal division ratio kappa = x/r of a segment AB to the Apollonian distance ratio k = (1 + kappa)/(1 - kappa).

F.16 d’Alembert’s Theorem

Given three circles of distinct radii in the plane, their six pairwise internal and external centers of similitude lie by threes on four projective lines L_{+++}, L_{–+}, L_{-+-}, and L_{+–} (the Monge–d’Alembert similitude axes).

F.39 Pole and Polar with Respect to a Triangle

Given a triangle ABC and a point P with cevian traces A’, B’, C’, the intersections X = BC cap B’C’, Y = CA cap C’A’, and Z = AB cap A’B’ are collinear on the trilinear polar L of the pole P.