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SMT 2020 Geometry #7

Source:

August 9, 2023
geometry

Problem Statement

Let ABCABC be an acute triangle with BC=4BC = 4 and AC=5AC = 5. Let DD be the midpoint of BCBC, EE be the foot of the altitude from BB to ACAC, and FF be the intersection of the angle bisector of ∠BCA\angle BCA with segment ABAB. Given that ADAD, BEBE, and CFCF meet at a single point PP, compute the area of triangle ABCABC. Express your answer as a common fraction in simplest radical form.
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