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6 planes are concurrent

Source: Romania BMO TST 1990 p4

February 19, 2020
3D geometrytetrahedronconcurrent planesincenterplanesconcurrent

Problem Statement

Let MM be a point on the edge CDCD of a tetrahedron ABCDABCD such that the tetrahedra ABCMABCM and ABDMABDM have the same total areas. We denote by πAB\pi_{AB} the plane ABMABM. Planes πAC,...,πCD\pi_{AC},...,\pi_{CD} are analogously defined. Prove that the six planes πAB,...,πCD\pi_{AB},...,\pi_{CD} are concurrent in a certain point NN, and show that NN is symmetric to the incenter II with respect to the barycenter GG.
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