Let's construct a triangle, given one vertex, the orthocenter, and the circumcenter of the triangle.
Let the given vertex be $A$, the orthocenter be $M$, and the circumcenter be $O$.

It is known that the reflections of the orthocenter over the sides of the triangle lie on the circumcircle (K. M. L. IX. 124.). Therefore, we draw the circumcircle of the triangle with the distance $AO$ as the radius. We draw a line through points $A$ and $M$, which intersects the circumference of the circle at point $N$. We erect a perpendicular at the midpoint of segment $MN$, which intersects the circle at the desired points $B$ and $C$.

(Koffler Béla, Budapest.)