Problem A. 845. (February 2023)

A. 845. The incircle of triangle \(\displaystyle ABC\) is tangent to sides \(\displaystyle BC\), \(\displaystyle AC\), and \(\displaystyle AB\) at points \(\displaystyle D\), \(\displaystyle E\), and \(\displaystyle F\), respectively. Let \(\displaystyle A'\) denote the point of the incircle for which circle \(\displaystyle (A'BC)\) is tangent to the incircle. Define points \(\displaystyle B'\) and \(\displaystyle C'\) similarly. Prove that lines \(\displaystyle A'D\), \(\displaystyle B'E\) and \(\displaystyle C'F\) are concurrent.

Proposed by Áron Bán-Szabó, Budapest

(7 pont)

Deadline expired on March 10, 2023.

Let us draw the common tangent of the incircle and circle \(\displaystyle A'BC\) at point \(\displaystyle A'\), and let this line intersect the line of side \(\displaystyle BC\) at point \(\displaystyle X\), We will prove that \(\displaystyle X\) is on the radical axis of the incircle and the circumcircle of triangle \(\displaystyle ABC\). Indeed: \(\displaystyle X\) is actually the radical center of the incircle, the circucircle and circle \(\displaystyle A'BC\), since \(\displaystyle BC\) is the radical axis of the circumcircle and circle \(\displaystyle A'BC\), while line \(\displaystyle XA'\) is the radical axis of the incircle and circle \(\displaystyle A'BC\). Similarly defining points \(\displaystyle Y\) and \(\displaystyle Z\), they are also on the radical axis of the incircle and the circumcircle. Now observe that line \(\displaystyle A'D\) is the polar of point \(\displaystyle X\) with respect to the incircle, \(\displaystyle B'E\) is the polar of \(\displaystyle Y\) and \(\displaystyle C'F\) is the polar of \(\displaystyle Z\), thus \(\displaystyle A'D\), \(\displaystyle B'E\) and \(\displaystyle C'F\) all pass through the pole of line \(\displaystyle XYZ\) with respect to the incircle.

Statistics:

17 students sent a solution.
7 points:	Bodor Mátyás, Chrobák Gergő, Diaconescu Tashi, Foris Dávid, Lovas Márton, Nádor Benedek, Seres-Szabó Márton, Sida Li, Szakács Ábel, Sztranyák Gabriella, Tarján Bernát, Varga Boldizsár, Virág Rudolf.
6 points:	Móricz Benjámin, Németh Márton, Simon László Bence, Wiener Anna.

Problems in Mathematics of KöMaL, February 2023