Question 120
Consider an acute triangle $A B C$ and squares $A B Z X$ and $A C W Y$ constructed externally on its sides. The points $M$ and $P$ are the midpoints of the segments $B C$ and $X Y$, respectively; the points $Q$ and $N$ are the centers of the squares $A B Z X$ and $A C W Y$, respectively; and the points $R$ and $S$ are the midpoints of the sides $A B$ and $A C$, respectively.
We will prove that $M N P Q$ is a square in a few steps.
a) Verify that triangles $Q R M$ and $M S N$ are congruent.
b) Verify that $\angle Q M N=90^{\circ}$.
c) Knowing the previous results, show that $M N P Q$ is a square.
We will prove that $M N P Q$ is a square in a few steps.
a) Verify that triangles $Q R M$ and $M S N$ are congruent.
b) Verify that $\angle Q M N=90^{\circ}$.
c) Knowing the previous results, show that $M N P Q$ is a square.