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points in space with the same locus, circumcenter of a tetrahedron related

Source: Vietnamese MO (VMO) 1965

August 22, 2018
Locus3-Dimensional Geometrygeometry3D geometrytetrahedron

Problem Statement

ABAB and CDCD are two fixed parallel chords of the circle SS. MM is a variable point on the circle. QQ is the intersection of the lines MDMD and ABAB. XX is the circumcenter of the triangle MCQMCQ. Find the locus of XX. What happens to XX as MM tends to (1) DD, (2) CC? Find a point EE outside the plane of SS such that the circumcenter of the tetrahedron MCQEMCQE has the same locus as XX.
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