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Girls in Math at Yale 2022 Problem 8: Maximize [DEF]^2

Source:

February 27, 2022
Yalecollege

Problem Statement

Triangle ABCABC has sidelengths AB=1AB=1, BC=3BC=\sqrt{3}, and AC=2AC=2. Points D,ED,E, and FF are chosen on AB,BCAB, BC, and ACAC respectively, such that ∠EDF=∠DFA=90∘\angle EDF = \angle DFA = 90^{\circ}. Given that the maximum possible value of [DEF]2[DEF]^2 can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd⁡(a,b)=1\gcd (a, b) = 1, find a+ba + b. (Here [DEF][DEF] denotes the area of triangle DEFDEF.)
Proposed by Vismay Sharan
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