# Embarrassing to the Solver

Geometry Level 5

In triangle $$ABC$$, the angle bisector from $$A$$ and the perpendicular bisector of $$BC$$ meet at point $$D$$, the angle bisector from $$B$$ and the perpendicular bisector of $$AC$$ meet at point $$E$$, and the perpendicular bisectors of $$BC$$ and $$AC$$ meet at point F. Given that $$\angle ADF = 5$$ , $$\angle BEF = 10$$ , and $$AC = 3$$, let the length of $$DF$$ be $$\sqrt{a}$$ where $$a$$ is a square-free positive integer. Find $$a$$.

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