An invariant area ratio for generalized rectangle flanks of a triangle.

Source abstract

We present a one-parameter generalization of rectangle flanks of a triangle. Given an arbitrary triangle and a positive real number k, first-order rectangle flanks are constructed externally on the three sides so that each flank side is k times the corresponding free side. The outer free sides determine three associated triangle flanks, on which second-order rectangle flanks are constructed with the same parameter. We prove that the ratio of the total area of the second-order rectangle flanks to the total area of the first-order rectangle flanks is 3/k^2, independently of the shape of the initial triangle. This result unifies the two previously known cases k=1 and k=2.