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tetrahedron with AB x CD = AC x BD = AD x BC

Source: Polish MO second round 1956 p6

August 29, 2024
3D geometrygeometrytetrahedron

Problem Statement

Prove that if in a tetrahedron ABCD ABCD the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point, then
AB⋅CD=AC⋅BD=AD⋅BCAB \cdot CD = AC \cdot BD = AD \cdot BC and that the converse also holds.
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