Mathematical and Physical Journal
for High Schools
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# Problem A. 754. (May 2019)

A. 754. Let $\displaystyle P$ be a point inside the acute triangle $\displaystyle ABC$, and let $\displaystyle Q$ be the isogonal conjugate of $\displaystyle P$. Let $\displaystyle L$, $\displaystyle M$ and $\displaystyle N$ be the midpoints of the shorter arcs $\displaystyle BC$, $\displaystyle CA$ and $\displaystyle AB$ of the circumcircle of $\displaystyle ABC$, respectively. Let $\displaystyle X_A$ be the intersection of ray $\displaystyle LQ$ and circle $\displaystyle PBC$, let $\displaystyle X_B$ be the intersection of ray $\displaystyle MQ$ and circle $\displaystyle PCA$, and let $\displaystyle X_C$ be the intersection of ray $\displaystyle NQ$ and circle $\displaystyle PAB$. Prove that $\displaystyle P$, $\displaystyle X_A$, $\displaystyle X_B$ and $\displaystyle X_C$ are concyclic or coincide.

Proposed by: Gustavo Cruz (São Paulo)

(7 pont)

Deadline expired on June 11, 2019.

### Statistics:

 3 students sent a solution. 7 points: Schrettner Jakab. 3 points: 1 student. 0 point: 1 student.

Problems in Mathematics of KöMaL, May 2019