Flying In The Middle Of Nowhere

Geometry Level 5

In the diagram, triangle \(ABC\) inscribed a circle with center \(O\).

Draw altitudes \(AD, BE, CF\) of the triangle \(ABC\) (\(D, E, F\) lies on the sides). The altitudes concur at orthocenter \(H\).

\(DE\) intersects \(CF\) at \(M\). \(DF\) intersects \(BE\) at \(N\).

A line passes through \(A\) and perpendicular to \(MN\) intersects \(OH\) at point \(K\).

If \(S_{ABC} = 2016^{2017}, P_{DEF} = 1344^{2016}\), length of \(KD\) can be written as:

\[KD=\dfrac{p_1^{\alpha_1} \times p_2^{\alpha_2}}{p_3^{\alpha_3}}\]

where \(p_1, p_2\) and \( p_3\) are 3 distinct primes.

Find \(\displaystyle \sum_{x=1}^3 (p_x\alpha_x)\).

Clarification: \(S\) and \( P\) denote the area and perimeter of the figure, respectively.


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