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Real IMO Shortlist 2018 G5(FL passes through centre of insphere of tetrahedron)

Source: https://artofproblemsolving.com/community/c6h1676184p10677746

March 27, 2020
3D geometrygeometryspheretetrahedroncollinear

Problem Statement

A sphere SS lies within tetrahedron ABCDABCD, touching faces ABD,ACDABD, ACD, and BCDBCD, but having no point in common with plane ABCABC. Let EE be the point in the interior of the tetrahedron for which SS touches planes ABEABE, ACEACE, and BCEBCE as well. Suppose the line DEDE meets face ABCABC at FF, and let LL be the point of SS nearest to plane ABCABC. Show that segment FLFL passes through the centre of the inscribed sphere of tetrahedron ABCEABCE.
KöMaL A.723. (April 2018), G. Kós
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