[math]M[/math] is the midpoint of side [math]\overline{PQ}[/math] of [math]\triangle{PQR}[/math]. Any line is drawn through [math]R[/math] outside the triangle and perpendiculars [math]\overline{PX}[/math], [math]\overline{MY}[/math], and [math]\overline{QZ}[/math] are drawn to this line. Prove that [math]\overline{XM}\cong\overline{ZM}[/math].

In the above diagram, you can move point [math]R[/math] to change what the original triangle looks like.[br]You can also move [math]S[/math] around to change which line through [math]R[/math]. Notice that the blue segments always appear to be congruent