Triangle with an Altitude

Geometry Level 4

\(ABC\) is a right triangle with \(\angle ABC = 90^\circ\). Point \(D\) is the midpoint of \(BC\), and point \(E\) is the foot of the perpendicular from \(B\) to \(AC\). \(AD\) and \(BE\) intersect at \(F\). If \(AB = 36\) and \(\angle BAD = \angle BCA\), what is the value of \(FE^2\)?

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