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Equivalence of identities in a triangle

Source: Turkey JBMO TST 2014 P7

June 21, 2014
geometryincenterangle bisectorgeometry proposed

Problem Statement

Let a line ℓ\ell intersect the line ABAB at FF, the sides ACAC and BCBC of a triangle ABCABC at DD and EE, respectively and the internal bisector of the angle BACBAC at PP. Suppose that FF is at the opposite side of AA with respect to the line BCBC, CD=CECD = CE and PP is in the interior the triangle ABCABC. Prove that FB⋅FA+CP2=CF2  ⟺  AD⋅BE=PD2.FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.
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