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.


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

Problems in Mathematics of KöMaL, May 2019