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special tetrahedron; prove that S'A' = S'B' = S'C'

Source: All-Russian Olympiad 2006 finals, problem 11.6

May 6, 2006
geometry3D geometrytetrahedronsphereradical axispower of a pointgeometry proposed

Problem Statement

Consider a tetrahedron SABCSABC. The incircle of the triangle ABCABC has the center II and touches its sides BCBC, CACA, ABAB at the points EE, FF, DD, respectively. Let A′A^{\prime}, B′B^{\prime}, C′C^{\prime} be the points on the segments SASA, SBSB, SCSC such that AA′=ADAA^{\prime}=AD, BB′=BEBB^{\prime}=BE, CC′=CFCC^{\prime}=CF, and let S′S^{\prime} be the point diametrically opposite to the point SS on the circumsphere of the tetrahedron SABCSABC. Assume that the line SISI is an altitude of the tetrahedron SABCSABC. Show that S′A′=S′B′=S′C′S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}.
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