Math Gold Medalist

Lor

2023 AIME I 

Problem 12

Let $\triangle ABC$ be an equilateral triangle with side length $55.$ Points $D,$ $E,$ and $F$ lie on $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, with $BD = 7,$ $CE=30,$ and $AF=40.$ Point $P$ inside $\triangle ABC$ has the property that\[\angle AEP = \angle BFP = \angle CDP.\]Find $\tan^2(\angle AEP).$

Drawing Perpendiculars

Properties of a Point Inside an Equilateral Triangle

Properties of Fractions

Draw three perpendiculars from P to BC, AC, AB and call the feet X, Y, Z respectively.

tan(<AEP) = PX/DX = PY/YE = PZ/ZF

So tan(<AEP) = (PX+PY+PZ)/(DX+YE+ZF)

Prove that PX+PY+PZ = altitude of ABC

Prove that BX+CY+AZ = XC+YA+BZ

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   Solution