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fixed sum MM_1 + MM_2 + MM_3 in tetrahedron

Source: 1996 Romania NMO X p2

August 13, 2024
geometry3D geometryfixedtetrahedron

Problem Statement

Let ABCDABCD a tetrahedron and MM a variable point on the face BCDBCD. The line perpendicular to (BCD)(BCD) in MM . intersects the planes(ABC) (ABC), (ACD)(ACD), and (ADB)(ADB) in M1M_1, M2M_2, and M3M_3. Show that the sum MM1+MM2+MM3MM_1 + MM_2 + MM_3 is constant if and only if the perpendicular dropped from AA to (BCD)(BCD) passes through the centroid of triangle BCDBCD.
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