Question 5
Let $ABC$ be a triangle.
Points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$$\angle AFE = \angle BDF = \angle CED.$$
Let $O_A$, $O_B$, and $O_C$ be the circumcenters of triangles $AFE$, $BDF$, and $CED$, respectively.
Let $M$, $N$, and $O$ be the circumcenters of triangles $ABC$, $DEF$, and $O_AO_BO_C$, respectively.
Prove that $OM = ON$.
Proposed by Serena An.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$$\angle AFE = \angle BDF = \angle CED.$$
Let $O_A$, $O_B$, and $O_C$ be the circumcenters of triangles $AFE$, $BDF$, and $CED$, respectively.
Let $M$, $N$, and $O$ be the circumcenters of triangles $ABC$, $DEF$, and $O_AO_BO_C$, respectively.
Prove that $OM = ON$.
Proposed by Serena An.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.