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Israel 2015 Q5 - Tetrahedron and inscribed spheres

Source: Israel National Olympiad 2015 Q5

August 7, 2019
geometry3D geometrytetrahedronsphere

Problem Statement

Let ABCDABCD be a tetrahedron. Denote by S1S_1 the inscribed sphere inside it, which is tangent to all four faces. Denote by S2S_2 the outer escribed sphere outside ABCABC, tangent to face ABCABC and to the planes containing faces ABD,ACD,BCDABD,ACD,BCD. Let KK be the tangency point of S1S_1 to the face ABCABC, and let LL be the tangency point of S2S_2 to the face ABCABC. Let TT be the foot of the perpendicular from DD to the face ABCABC.
Prove that L,T,KL,T,K lie on one line.
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