Place three points A,B,C in the plane, not in a line. Put another point P strictly inside the triangle. Draw circles A,P,B; B,P,C; and C,P,A. Show that these three circles have no interior points in common, that is, their intersection is null (excluding the circumferences). This seems obvious but I need a geometric or algebraic proof and don't have one. (If A,B,C,P is convex the circles do have interior points in common.) Thank you for any hints or solutions.