BALTIC-WAY 2018 Problem 15

Two circles in the plane do not intersect and do not lie inside each other. We choose diameters $A_{1} B_{1}$ and $A_{2} B_{2}$ of these circles such that the segments $A_{1} A_{2}$ and $B_{1} B_{2}$ intersect. Let $A$ and $B$ be the midpoints of the segments $A_{1} A_{2}$ and $B_{1} B_{2}$, and $C$ be the intersection point of these segments. Prove that the orthocenter of the triangle $A B C$ belongs to a fixed line that does not depend on the choice of the diameters.