11.4. Given a triangle $A B C$. Where in its plane should a point $M$ be chosen so that the sum of the radii of the circumcircles of triangles $A B M$ and $B C M$ is minimized?
11.4. Answer. Point $M$ is the foot of the altitude $B M$ of triangle $A B C$. Notice that the diameters of the circles mentioned in the condition cannot be less than the chords $A B$ and $B C$ respectively.