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Show that R / r > a / h in any triangular pyramid ABCD

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 1

June 15, 2011
geometry3D geometrypyramidspheregeometry unsolved

Problem Statement

A triangular pyramid ABCDABCD is given. Prove that Rr>ah\frac Rr > \frac ah, where RR is the radius of the circumscribed sphere, rr is the radius of the inscribed sphere, aa is the length of the longest edge, hh is the length of the shortest altitude (from a vertex to the opposite face).