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Prove that A, B, C, D lie on the same line [ILL 1974]

Source:

January 2, 2011
geometry

Problem Statement

Let A,B,C,DA,B,C,D be points in space. If for every point MM on the segment ABAB the sum SAMC+SCMD+SDMBS_{AMC}+S_{CMD}+S_{DMB} Is constant show that the points A,B,C,DA,B,C,D lie in the same plane.
[hide="Note."] Note. SXS_X denotes the area of triangle X.X.
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