If p₁, p₂, p₃ are altitudes of a triangle A B C from the vertices A, B, C respectively and if is the area of the triangle, S is the semi perimeter of the triangle, then Cos A p ₁ + Cos B p ₂ + Cos C p ₃

Step-by-step solution

The area of a triangle can be expressed in terms of its altitudes p₁ , p₂ , p₃ and corresponding sides as = 1 2 a p₁ = 1 2 b p₂ = 1 2 c p₃ . Solving for the altitudes gives p₁ = 2 a , p₂ = 2 b , and p₃ = 2 c . Substituting into the expression E = A p₁ + B p₂ + C p₃ yields: E = A 2 a + B 2 b + C 2 c = 1 2 (a A + b B + c C) . Using the law of sines, a = 2R A , b = 2R B , and c = 2R C , the expression becomes: a A + b B + c C = 2R( A A + B B + C C) . Applying the identity 2x = 2 x x gives: 2R( A A + B B + C C) = R( 2A