Question 77
In a non-isosceles triangle $ABC$ let $O$ and $I$ be the circumcenter and the incenter, respectively. Let $D, E, F$ be the midpoints of the sides $[BC], [AC], [AB]$, respectively. Let $T$ be the foot of the perpendicular from $I$ to $[AB]$, $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of the line segment $[OI]$. If $A, P$ and $Q$ are collinear, prove that
$$
\frac{|AO|}{|OD|} - \frac{|BC|}{|AT|} = 4.
$$
$$
\frac{|AO|}{|OD|} - \frac{|BC|}{|AT|} = 4.
$$