A Problem in Elementary Geometry

Extract

Recently, in connection with some work on Diophantine approximations, I encountered the following problem on triangles.

Let T be a triangle with vertices A, B, C which are, respectively, inner points of the sides a, b, c of a second triangle t. Is it always possible to move T into a new position where its vertices are inner points of t?

I give here an affirmative answer to the problem and prove, moreover, that it suffices to apply to T an arbitrarily small rotation about a suitably chosen point of the plane. I am indebted to my Manchester colleagues for a number of simplifications of this solution, arrived at when discussing the problem with them.