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many points on a sphere :|

Source: bmo 1986

April 23, 2007
geometry3D geometryspheretetrahedronsearchcircumcirclegeometry proposed

Problem Statement

Let ABCDABCD be a tetrahedron and let E,F,G,H,K,LE,F,G,H,K,L be points lying on the edges AB,BC,CDAB,BC,CD ,DA,DB,DC,DA,DB,DC respectively, in such a way that AE⋅BE=BF⋅CF=CG⋅AG=DH⋅AH=DK⋅BK=DL⋅CL.AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL. Prove that the points E,F,G,H,K,LE,F,G,H,K,L all lie on a sphere.
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