Playing with Triangles!

Geometry Level 5

In a \(\Delta ABC\), let \(AB = AC = 5, \ BC = 6\). Let \(E\) be a point on \(AC\) and \(F\) be a point on \(AB\) such that \(BE=CF, \ \angle EBC \neq \angle FCB\) and \(\sin(\theta) = \frac{5}{13} \), where \(\theta = \angle EBC\). Let \(H\) be the point of intersection of \(BE\) and \(CF\), and let \(K\) be a point on \(BC\) such that \(HK\) is perpendicular to \(BC\).

If the length of \(HK\) can be represented as \(\dfrac{\alpha}{\beta}\) where \(\alpha,\beta \in \mathbb Z^+, \ \ \gcd(\alpha,\beta) = 1 \), submit the value of \(\alpha + \beta\) as your answer.

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