Question 40
Every triangle in the plane $\mathbb{R}^2$ contains a Mediterranean point on its boundary or in its interior (even if the triangle degenerates into a line segment or a point). These Mediterranean points satisfy the following conditions:
(i) If a triangle is symmetric with respect to a line through the origin $(0, 0)$, then its Mediterranean point lies on this line.
(ii) If a triangle $DEF$ contains a triangle $ABC$, and if $ABC$ contains the Mediterranean point $M$ of $DEF$, then $M$ is also the Mediterranean point of $ABC$.
Determine all possible locations for the Mediterranean point of the triangle with vertices $(-3, 5)$, $(12, 5)$, and $(3, 11)$.
(i) If a triangle is symmetric with respect to a line through the origin $(0, 0)$, then its Mediterranean point lies on this line.
(ii) If a triangle $DEF$ contains a triangle $ABC$, and if $ABC$ contains the Mediterranean point $M$ of $DEF$, then $M$ is also the Mediterranean point of $ABC$.
Determine all possible locations for the Mediterranean point of the triangle with vertices $(-3, 5)$, $(12, 5)$, and $(3, 11)$.