KöMaL problem A, 2018

Geometry Difficulty 8.0 Shortlist Prove it Hungary

A sphere $\mathcal{G}$ lies within tetrahedron $ABCD$ , touching faces $ABD$ , $ACD$ , and $BCD$ , but having no point in common with plane $ABC$ . Let $E$ be the point in the interior of the tetrahedron for which $\mathcal{G}$ touches planes $ABE$ , $ACE$ , and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$ , and let $L$ be the point of $\mathcal{G}$ nearest to plane $ABC$ . Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$ .

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