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inscribed tetrahedra with orthogonal segments

Source: France 1991 P3

May 14, 2021
3D geometrygeometry

Problem Statement

Let SS be a fixed point on a sphere Σ\Sigma with center Ω\Omega. Consider all tetrahedra SABCSABC inscribed in Σ\Sigma such that SA,SB,SCSA,SB,SC are pairwise orthogonal. (a) Prove that all the planes ABCABC pass through a single point. (b) In one such tetrahedron, HH and OO are the orthogonal projections of SS and Ω\Omega onto the plane ABCABC, respectively. Let RR denote the circumradius of △ABC\triangle ABC. Prove that R2=OH2+2SH2R^2=OH^2+2SH^2.
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