(solved by Gaspard Ferey). Let $A$ and $B$ be two points in the plane and (d) a line that does not intersect the segment $[A B]$. Determine (geometrically) the point $M$ on (d) for which the angle $\widehat{A M B}$ is maximal.
. Let $M$ be any point in the plane for now and $R$ the radius of the circumcircle of triangle $A M B$. The formula $A B=2 R \sin \widehat{A M B}$ shows that the angle $\widehat{A M B}$ is maximal when $R$ is minimal (since $A B$ remains constant). Thus, it is a matter of finding the circle with the smallest radius passing through $A$ and $B$ and intersecting the line $(d)$, and it is obviously obtained when it is tangent to this line. The point $M$ sought is the point of tangency.