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vol KOAC/vol KOBD = AC/BD iff 2AC·BD = AB^2 in a tetrahedron ABCD

Source: Vietnamese MO (VMO) 1975 P3

August 20, 2018
geometry3D geometrytetrahedron

Problem Statement

Let ABCDABCD be a tetrahedron with BA⊥AC,DB⊥(BAC)BA \perp AC,DB \perp (BAC). Denote by OO the midpoint of ABAB, and KK the foot of the perpendicular from OO to DCDC. Suppose that AC=BDAC = BD. Prove that VKOACVKOBD=ACBD\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD} if and only if 2AC⋅BD=AB22AC \cdot BD = AB^2.
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