MathDB

Given a tetrahedron

A_1A_2A_3A_4

, define an

A_1

-exsphere such a sphere that is tangent to all planes given by faces of the tetrahedron and the vertex

A_1

and the sphere are separated by the plane

A_2A_3A_4.

Denote

\varrho_1,\ldots,\varrho_4

of all four exspheres. Furthermore, denote

v_i, i=1,\ldots,4

the distance of the vertex

A_i

from the opposite face. Show that

2\left(\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}+\frac{1}{v_4}\right)=\frac{1}{\varrho_1}+\frac{1}{\varrho_2}+\frac{1}{\varrho_3}+\frac{1}{\varrho_4}.

Problem Statement