Question 96
Let $ABC$ be a triangle and $M$ a point in its plane, distinct from $A$, $B$ and $C$. Let $N$, $P$ and $Q$ denote the symmetries of point $M$ with respect to sides $AB$, $BC$ and $AC$, respectively.
a) Prove that the points $N$, $P$ and $Q$ are collinear if and only if the point $M$ belongs to the circumcircle of triangle $ABC$.
b) If point $M$ does not belong to the circumcircle of triangle $ABC$ and triangles $ABC$ and $NPQ$ have the same centroid, prove that triangle $ABC$ is equilateral.
a) Prove that the points $N$, $P$ and $Q$ are collinear if and only if the point $M$ belongs to the circumcircle of triangle $ABC$.
b) If point $M$ does not belong to the circumcircle of triangle $ABC$ and triangles $ABC$ and $NPQ$ have the same centroid, prove that triangle $ABC$ is equilateral.