Have been reading Shwarz's proof of the minimal permiter of a triangle inscribed in another triangle as outlined in chapter VII of Courant and Robbins' "What is Mathematics" and they assume (on pg. 349 of second edition) that the minimal perimeter provided by "altitude" inscribed triangle construction would also be minimal on the sum of any two of its sides (let's call each of these pairwise sums the "partial perimeter" and the set of all three such sums the "partial perimeters").
I don't see how this can be assumed. Does anyone have a counterexample or demonstration that the inscribed triangle with the smallest permieter will also have the smallest sum of any two of its sides? If the inscribed triangle of minimal perimeter does not minimize the partial permiters, then can somebody characterize the inscribed triangle that provides a minimal partial perimeter?