Question 33
Let $X$ be an arbitrary point on the circumcircle of the triangle $ABC$. The perpendicular lines from $X$ to $AB$ and $AC$ cut $BC$ at $P$ and $Q$ respectively. The point $Y$ is the circumcenter of the circle passing through $X, P, Q$. (If $X, P, Q$ are coincident, then consider $Y$ the same point as them.)
a. Prove that if the triangle $ABC$ is equilateral, by moving $X$ on circumcircle, $Y$ moves on a circle.
b. Prove the converse of the previous part: If by moving $X$ on the circumcircle, $Y$ moves on a circle, then $ABC$ is equilateral.
a. Prove that if the triangle $ABC$ is equilateral, by moving $X$ on circumcircle, $Y$ moves on a circle.
b. Prove the converse of the previous part: If by moving $X$ on the circumcircle, $Y$ moves on a circle, then $ABC$ is equilateral.