Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 724. (April 2018)

A. 724. A sphere $\displaystyle \mathcal{G}$ lies within tetrahedron $\displaystyle ABCD$, touching faces $\displaystyle ABD$, $\displaystyle ACD$, and $\displaystyle BCD$, but having no point in common with plane $\displaystyle ABC$. Let $\displaystyle E$ be the point in the interior of the tetrahedron for which $\displaystyle \mathcal{G}$ touches planes $\displaystyle ABE$, $\displaystyle ACE$, and $\displaystyle BCE$ as well. Suppose the line $\displaystyle DE$ meets face $\displaystyle ABC$ at $\displaystyle F$, and let $\displaystyle L$ be the point of $\displaystyle \mathcal{G}$ nearest to plane $\displaystyle ABC$. Show that segment $\displaystyle FL$ passes through the centre of the inscribed sphere of tetrahedron $\displaystyle ABCE$.

(5 pont)

Deadline expired on May 10, 2018.

### Statistics:

 1 student sent a solution. 2 points: 1 student.

Problems in Mathematics of KöMaL, April 2018