Problem A. 828. (May 2022)

A. 828. Triangle \(\displaystyle ABC\) has incenter \(\displaystyle I\) and excircles \(\displaystyle \Omega_A\), \(\displaystyle \Omega_B\), and \(\displaystyle \Omega_C\). Let \(\displaystyle \ell_A\) be the line through the feet of the tangents from \(\displaystyle I\) to \(\displaystyle \Omega_A\), and define lines \(\displaystyle \ell_B\) and \(\displaystyle \ell_C\) similarly. Prove that the orthocenter of the triangle formed by lines \(\displaystyle \ell_A\), \(\displaystyle \ell_B\), and \(\displaystyle \ell_C\) coincides with the Nagel point of triangle \(\displaystyle ABC\).

(The Nagel point of triangle \(\displaystyle ABC\) is the intersection of segments \(\displaystyle AT_A\), \(\displaystyle BT_B\), and \(\displaystyle CT_C\), where \(\displaystyle T_A\) is the tangency point of \(\displaystyle \Omega_A\) with side \(\displaystyle BC\), and points \(\displaystyle T_B\) and \(\displaystyle T_C\) are defined similarly.)

Proposed by Nikolai Beluhov, Bulgaria

(7 pont)

Deadline expired on June 10, 2022.

Solution of the problem's proposer: three-polars-1.pdf

Statistics:

4 students sent a solution.
7 points:	Diaconescu Tashi, Seres-Szabó Márton, Sztranyák Gabriella.
6 points:	Ho Tran Khanh Linh.

Problems in Mathematics of KöMaL, May 2022