Find the area of a triangle if its sides are 7:15:20 apart and the circumcircle has a radius of 25.

Let x be the coefficient of proportionality, then the sides of the triangle are equal to a = 7x, b = 15x, c = 20x.
The radius of the circumscribed circle is found by the formula:
R = abc / 4S,
where S is the area of a triangle inscribed in a circle.
We find the area of the triangle using Heron’s formula:
S = √p (p – a) (p – b) (p – c),
where p is a semi-perimeter.
p = (a + b + c) / 2;
p = (7x + 15x + 20x) / 2 = 42x / 2 = 21x.
S = √21x (21x – 7x) (21x – 15x) (21x – 20x) = √ (21x * 14x * 6x * x) = √ (1764x ^ 4) = 42x ^ 2.
Then:
25 = (7x * 15x * 20x) / 4 * 42x ^ 2;
25 = 2100x ^ 3 / 168x ^ 2;
25 = 25x ^ 3 / 2x ^ 2;
25 = 25x / 2;
25x = 25 * 2 (proportional);
25x = 50;
x = 50/25;
x = 2.
Thus:
S = 42x ^ 2 = 42 * 2 ^ 2 = 42 * 4 = 168.
Answer: S = 168.