Maximizing sums of distances - Colegiul Național din Iași (Iași - Roumanie) Collège Sainte Véronique (Liège)

Colegiul Național din Iași (Iași - Roumanie)
Let ABCD be a rectangle and M and N two fixed points in its interior. Find the point P in the rectangle ABCD with the property that the sum PM+PN is as large as possible. Is it true that P is always one of the corners of the rectangle?
Generalizations: What if we replace ABCD by another shape (a circle, a triangle, ...)? What if there are more fixed points (e.g. A, B, C) and we want to maximize the sum PA+PB+PC?
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