In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x² - c² = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

Given a+b=x and ab=y
If x² - c² = y ⇒ (a+b)² - c² = ab ⇒ a² + b² - c² = -ab ⇒ a² + b² + 2ab - c² = 3ab
⇒ (a+b)² - c² = 3ab ⇒ x² - c² = 3y
Using cosine rule:
cosC = (a² + b² - c²) / 2ab = (x² - c² - 2ab) / 2y = (3y - 2y) / 2y = ½
Therefore, angle C = 60°
Circumradius R = c / 2sinC = c / 2sin60° = c / 2(√3/2) = c/√3 = c√3