Question 124
Consider a triangle $ABC$ with $\angle BAC = 120^\circ$ and the isosceles triangles $PAB$ and $NAC$ such that $\angle APB = \angle ANC = \angle BAC$, with line $AB$ separating points $P$ and $C$, and line $AC$ separating points $N$ and $B$. Prove that, if $G$ is the centroid of triangle $ABC$, then $GP = GN = \frac{AB+AC}{3}$.