Let $C_1$ and $C_2$ be two circles in the plane with radius R and 3R respectively. Show that every point in the interior of $C_2$ is the orthocentre of some triangle inscribed in $C_1$. I gave a construction as follows. Take any point, call it H in the interior of $C_2$. Join OH, it intersects the circle $C_1$ at two points say $A$ and $X$ with $A$ being nearer to $H$. Construct perpendicular bisector of $AX$. Let it intersect $C_1$ at $B$ and $C$. I tell that $ABC$ is the required triangle.
If I assume $H$ to be the orthocentre then all the properties are matching. However, I am unable to prove that the above construction will guarantee that H will be the orthocentre of triangle ABC.
Any help will be appreciated. Thanks in advance