Question 52
In the plane there are a triangle $\triangle ABC$ and a point $O$; $X$ lies on the ray $AC$. Take the point $X_1$ on the ray $BA$ such that $AX = AX_1$ and $A$ lies between $B$ and $X_1$. Then take the point $X_2$ on the ray $BC$ such that $\overline{XX_2} \parallel \overline{OC}$.
Prove that, as $X$ moves along the ray $AC$, the locus of the circumcentre of $\triangle BX_1X_2$ is part of a straight line.
Prove that, as $X$ moves along the ray $AC$, the locus of the circumcentre of $\triangle BX_1X_2$ is part of a straight line.