An Inequality concerning Polyhedra

This is easily established, since according to the theorem of Euler mentioned above, if d denotes the distance between the centers of the circumscribed and inscribed circles, then d = R2rR. It follows that R~2rR^0, and therefore R^2r. Equality holds only if the two circles are concentric, that is, if the triangle is equilateral. The problem of generalizing the above result to space was proposed by Professor L. Fejér. A young mathematician, I. Âdâm, deported to Germany during the war—where all traces of him have been lost—found and communicated to Professor Fejér in 1943 a very simple proof of the above extremum property of the equilateral triangle. His proof, which may be immediately generalized to space, runs as follows: If p is the radius of the circle passing through the midpoints of the sides of the triangle, then p = i?/2, and all that need be shown is that p is at least equal to the radius of the inscribed circle. This follows from the fact that the inscribed circle is the smallest among all circles which have common points with all three sides of the triangle. Such a circle is, namely, the circle inscribed in a homothetic triangle containing the original one.