Problem 6 : Strange Fractions

Geometry Level 5

In a non-isoceles 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 perpendicular from \(I\) to \(AB\), \(P\) be the circumcenter of triangle \(DEF\), and \(Q\) be the midpoint of \(OI\). If \(A, P, Q\) are collinear, the maximum possible value of \(|\frac{AO}{OD} - \frac{BC}{AT}|\) is \(\frac{a\sqrt{b}}{c}\), where \(b\) is not divisible by the square of any prime and \(a\) and \(c\) are relatively prime positive integers. Determine the value of \(a + b + c\).

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