Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.
Source: Mathematics Regional Olympiad of Mexico West 2017 P4
Tags: geometry, equal areas
Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.