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Arbitrary point P inside square ABCD

Source: Benelux MO 2014 Problem 4

July 17, 2014
inequalitiestrigonometrygeometry unsolvedgeometry

Problem Statement

Let ABCDABCD be a square. Consider a variable point PP inside the square for which ∠BAP≥60∘.\angle BAP \ge 60^\circ. Let QQ be the intersection of the line ADAD and the perpendicular to BPBP in PP. Let RR be the intersection of the line BQBQ and the perpendicular to BPBP from CC.
[*] (a) Prove that ∣BP∣≥∣BR∣|BP|\ge |BR|
[*] (b) For which point(s) PP does the inequality in (a) become an equality?
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