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IOQM P27 2024

Source:

September 8, 2024
geometry

Problem Statement

In a triangle ABCABC, a point PP in the interior of ABCABC is such that ∠BPC−∠BAC=∠CPA−∠CBA=∠APB−∠ACB. \angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB. Suppose ∠BAC=30∘\angle BAC = 30^{\circ} and AP=12AP = 12. Let D,E,FD,E,F be the feet of perpendiculars from PP on to BC,CA,ABBC,CA,AB respectively. If mnm \sqrt{n} is the area of the triangle DEF where m,nm,n are integers with nn prime, then what is the value of the product mnmn?
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