The sides of the triangle are 13, 14 and 15 cm. Find the ratio of the radii of the inscribed and circumscribed circles.

The radius of the inscribed and circumscribed circle can be found using the formula for the area of a triangle through the radius of the inscribed and circumscribed circle.

S = abc / (4R),

S = pr, where p = (a + b + c) / 2, where r and R are the radii of the inscribed and circumscribed circles.

Let’s express them from the formulas:

R = (a * b * c) / (4S), r = S / p

Let’s write the relation:

r / R = (4S ^ 2) / (p * a * b * c).

We find the area of the triangle using Heron’s formula:

p = (13 + 14 + 15) / 2 = 21

S = √ (p (p – a) (p – b) (p – c) = √ (21 * 8 * 7 * 6) = √7056 = 84.

r / R = (4 * 7056) / (21 * 13 * 14 * 15) = 32/65 (1: 2).

Answer: r / R = 32/65 (approximately 1: 2).