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The inscribed sphere of a tetrahedron

Source: Serbia and Montanagro 2003 \grade 11\second day

August 3, 2005
geometry3D geometryspheretetrahedroncircumcircleradical axisgeometry unsolved

Problem Statement

The inscribed sphere of a tetrahedron ABCDABCD touches ABC,ABD,ACDABC,ABD,ACD and BCDBCD at D1,C1,B1D_1,C_1,B_1 and A1A_1 respectively. Consider the plane equidistant from AA and plane B1C1D1B_1C_1D_1 (parallel to B1C1D1B_1C_1D_1) and the three planes defined analogously for the vertices B,C,DB,C,D. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of ABCDABCD.
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