Question 63
Let us consider a triangle with side lengths $a, b, c$ such that
$$
b(a+b)(b+c) = a^3 + b(a^2 + c^2) + c^3.
$$
Let us call $A, B$ and $C$ the values, measured in radians, of the angles of the triangle. Prove that the equality $\frac{1}{\sqrt{A}+\sqrt{B}} + \frac{1}{\sqrt{B}+\sqrt{C}} = \frac{2}{\sqrt{A}+\sqrt{C}}$ holds.
$$
b(a+b)(b+c) = a^3 + b(a^2 + c^2) + c^3.
$$
Let us call $A, B$ and $C$ the values, measured in radians, of the angles of the triangle. Prove that the equality $\frac{1}{\sqrt{A}+\sqrt{B}} + \frac{1}{\sqrt{B}+\sqrt{C}} = \frac{2}{\sqrt{A}+\sqrt{C}}$ holds.