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tetrahedron inequality V_1 +V_2 +V_3 \ge 3V

Source: Polish second round 1993 p3

January 19, 2020
geometry3D geometrytetrahedroninequalitiesgeometric inequality

Problem Statement

A tetrahedron OA1B1C1OA_1B_1C_1 is given. Let A2,A3∈OA1,A2,A3∈OA1,A2,A3∈OA1A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1 be points such that the planes A1B1C1,A2B2C2A_1B_1C_1,A_2B_2C_2 and A3B3C3A_3B_3C_3 are parallel and OA1>OA2>OA3>0OA_1 > OA_2 > OA_3 > 0. Let ViV_i be the volume of the tetrahedron OAiBiCiOA_iB_iC_i (i=1,2,3i = 1,2,3) and VV be the volume of OA1B2C3OA_1B_2C_3. Prove that V1+V2+V3≥3VV_1 +V_2 +V_3 \ge 3V.
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