CeviansAndParallels

Cevians and parallels

Consider a triangle ABC a point G, the Cevians of it, the cevian triangle DEF, the trilinear polar A'B' of G etc.. Then take paralles to the sides of the triangle in respective distances h₁, h₂, h₃. Find a necessary condition for the h_i, such that the intersection points {D',E',F'} of these parallels with the sides of DEF are again on a line.

After Menelaus a necessary and sufficient condition for the collinearity of {D',E',F'} is (F'D/F'E)*(D'E/D'F)*(E'F/E'D) = 1. Denoting by [D,AB] the distance of point D from line AB this amounts to the condition:

This property is used in EqualCirclesAtVertices.html to prove the perspectivity of two important triangles.

Cevians and parallels

See Also