Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $H$ be the orthocenter of $A B C$. Find the distance between the circumcenters of triangles $A H B$ and $A H C$.
Let $H_{B}$ be the reflection of $H$ over $A C$ and let $H_{C}$ be the reflection of $H$ over $A B$. The reflections of $H$ over $A B, A C$ lie on the circumcircle of triangle $A B C$. Since the circumcenters of triangles $A H_{C} B, A H_{B} C$ are both $O$, the circumcenters of $A H B, A H C$ are reflections of $O$ over $A B, A C$ respectively. Moreover, the lines from $O$ to the circumcenters in question are the perpendicular bisectors of $A B$ and $A C$. Now we see that the distance between the two circumcenters is simply twice the length of the midline of triangle $A B C$ that is parallel to $B C$, meaning the distance is $2\left(\frac{1}{2} B C\right)=14$. $\fbox{14}$.