# 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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