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.


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

Problems in Mathematics of KöMaL, April 2018