Problem

Source: Romanian JBTST II 2007, problem 3

Tags: trigonometry, inequalities, geometry, conics, ellipse, symmetry, perpendicular bisector



Let $ABC$ an isosceles triangle, $P$ a point belonging to its interior. Denote $M$, $N$ the intersection points of the circle $\mathcal{C}(A, AP)$ with the sides $AB$ and $AC$, respectively. Find the position of $P$ if $MN+BP+CP$ is minimum.