Junior Problems J493. in Triangle ABC, R = 4R. Prove That ∠A
Junior problems ○ J493. In triangle ABC, R = 4r. Prove that ∠A − ∠B = 90 if and only if ¾ ab a b c2 : − = − 2 Proposed by Adrian Andreescu, University of Texas at Austin, USA Solution by Daniel Lasaosa, Pamplona, Spain Note that the second condition is equivalent to 3ab 2 2 2 3 C 1 = a + b − c = 2ab cos C; cos C = ; sin = √ : 2 4 2 2 2 A B C Since it is well known that r = 4R sin 2 sin 2 sin 2 , the second condition in a triangle such that R = 4r also sin A sin B √1 gives 2 2 = 4 2 , hence A − B A + B A B C A B 1 ○ cos = cos + 2 sin sin = sin + 2 sin sin = √ = cos 45 ; 2 2 2 2 2 2 2 2 ○ ○ A B C which implies to ∠A − ∠B = 90 . Conversely, if ∠A − ∠B = 90 , then using R = 4r so 1 = 16 sin 2 sin 2 sin 2 we find 1 A B C A B C 1 cos − sin 2 sin sin sin : √ = 2 = 2 + 2 2 = 2 + C 2 8 sin 2 sin C sin C √1 This quadratic equation in 2 has a double root and hence implies 2 = 2 2 , which we saw above was equivalent to the second equation. The conclusion follows. Also solved by Ioannis D. Sfikas, Athens, Greece; Polyahedra, Polk State College, USA; Arkady Alt, San Jose, CA, USA; Nicusor Zlota, Traian Vuia Technical College, Focsani, Romania; Dumitru Barac, Sibiu, Romania; Marin Chirciu, Colegiul Nat,ional Zinca Golescu, Pites, ti, Romania; Nguyen Viet Hung, Hanoi University of Science, Vietnam; Titu Zvonaru, Comănes, ti, Romania; Taes Padhihary, Disha Delphi Public School, India; Vicente Vicario García, Sevilla.
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