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2

Part of 1973 Czech and Slovak Olympiad III A

Problems(1)

Ex-spheres of tetrahedron

Source: Czech and Slovak Olympiad 1973, National Round, Problem 2

7/16/2024
Given a tetrahedron A1A2A3A4A_1A_2A_3A_4, define an A1A_1-exsphere such a sphere that is tangent to all planes given by faces of the tetrahedron and the vertex A1A_1 and the sphere are separated by the plane A2A3A4.A_2A_3A_4. Denote ϱ1,…,ϱ4\varrho_1,\ldots,\varrho_4 of all four exspheres. Furthermore, denote vi,i=1,…,4v_i, i=1,\ldots,4 the distance of the vertex AiA_i from the opposite face. Show that 2(1v1+1v2+1v3+1v4)=1ϱ1+1ϱ2+1ϱ3+1ϱ4.2\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}.
geometry3D geometrytetrahedronspheretangent spheres