17/07/2024

Daca un triunghi ABC are laturile in progresie aritmetica cu ratia r raza cercului inscris, calculati cosinusurile unghiurilor.

Presupunem ca a = b – r si c = b + r
p = (a+b+c)/ 2 = 3b/2 -> semiperimetrul

Notam aria cu S. Conform formulei lui Heron:

S = √ p(p-a)(p-b)(p-c) = √ 3b/2(3b/2 – b+r)(3b/2 – b)(3b/2 – b-r) = √3b/4(b/2+r)(b/2-r) =>

S = b√3 / 2 (√ (b2/4 – r2)

Dar S = rp unde r = raza cerc inscris

Deci r2p2 = S2

9r2=3(b2/4 – r2) => 12r2 = 3b2/4 => 48 r2 = 3b2 => 16 r2 = b2 => b = 4r

Deci a = 3r, b = 4r, c = 5r

Conform reciprocei teoremei lui Pitagora, deoarece a2+b2 = c2 => triunghiul ABC este dreptunghic cu ipotenuza c => cos C = 0, cos B = 3/5, cos A = 4/5

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