Question 12
Point $M$ is marked inside the convex quadrilateral $ABCD$ so that the ratio of the areas of the triangles $AMC$ and $BMD$ is equal to the ratio of the tangents of the angles $AMC$ and $BMD$, i.e. $S(AMC) : S(BMD) = \tan \angle AMC : \tan \angle BMD$.
Prove that $AM^2 + MC^2 + BD^2 = AC^2 + BM^2 + MD^2$ if $M$ does not belong to any of the diagonals of the quadrilateral.
Prove that $AM^2 + MC^2 + BD^2 = AC^2 + BM^2 + MD^2$ if $M$ does not belong to any of the diagonals of the quadrilateral.