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Find the locus of the point P

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September 22, 2010
3D geometrytetrahedronsphereLocusLocus problemsIMO Shortlistgeometry

Problem Statement

Let a tetrahedron ABCDABCD be inscribed in a sphere SS. Find the locus of points PP inside the sphere SS for which the equality APPA1+BPPB1+CPPC1+DPPD1=4\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4 holds, where A1,B1,C1A_1,B_1, C_1, and D1D_1 are the intersection points of SS with the lines AP,BP,CPAP,BP,CP, and DPDP, respectively.
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