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M is interor point of ABCD - volume problem

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September 23, 2010
geometry3D geometrytetrahedronVolumeequationIMO Shortlist

Problem Statement

Let MM be an interior point of the tetrahedron ABCDABCD. Prove that  MA⟶vol(MBCD)+MB⟶vol(MACD)+MC⟶vol(MABD)+MD⟶vol(MABC)=0 \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array} (vol(PQRS)\text{vol}(PQRS) denotes the volume of the tetrahedron PQRSPQRS).
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