MathDB
medians, tetrahedron and inequality

Source: Polish MO 1997

July 31, 2005
geometry3D geometrytetrahedroninequalitiestrigonometry

Problem Statement

In a tetrahedron ABCDABCD, the medians of the faces ABDABD, ACDACD, BCDBCD from DD make equal angles with the corresponding edges ABAB, ACAC, BCBC. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces. [hide="Comment"]Equivalent version of the problem: ABCDABCD is a tetrahedron. DEDE, DFDF, DGDG are medians of triangles DBCDBC, DCADCA, DABDAB. The angles between DEDE and BCBC, between DFDF and CACA, and between DGDG and ABAB are equal. Show that: area DBCDBC ≤\leq area DCADCA + area DABDAB.
Back to ProblemsView on AoPS