Question 48
Let $ABC$ A acute triangle. be $M_{A}$, $M_{B}$ and $M_{C}$ the respective midpoints of the sides $BC$, $CA$ and $AB$. be $M_{A}^{\prime}$, $M_{B}^{\prime}$ and $M_{C}^{\prime}$ the respective backgrounds of the minor arcs $BC$, $CA$ and $AB$ on the circle circumscribed to the triangle $ABC$. Let $P_{A}$ the intersection of the line $M_{B}M_{C}$ and from the perpendicular to $M_{B}^{\prime}M_{C}^{\prime}$ by $A$. The points $P_{B}$ and $P_{C}$ are defined in a similar way. Show that the lines $M_{A}P_{A}$, $M_{B}P_{B}$ and $M_{C}P_{C}$ intersect at a point.