JEE Advance - Mathematics (1984 - No. 33)
With usual notation, if in a triangle $$ABC$$;
$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ then prove that $${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$$.
$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ then prove that $${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$$.
The given condition is incorrect and cannot be proved.
Let $b+c = 11k, c+a = 12k, a+b = 13k$. Solving this system of equations gives $a=7k, b=6k, c=5k$. Then use the cosine rule to calculate cos A, cos B, cos C.
Let $b+c = 11k, c+a = 12k, a+b = 13k$. Solving this system of equations gives $a=6k, b=7k, c=5k$. Then use the sine rule to calculate sin A, sin B, sin C.
The sines rule is enough for the solution.
The projection formulas are sufficient for solving this problem.