The center of the circumscribed circle lies at the height of an isosceles triangle and divides the height

The center of the circumscribed circle lies at the height of an isosceles triangle and divides the height into segments equal to 5cm and 13cm. Find the area of the triangle.

BH = BO + OH;
BH = 13 + 5;
BH = 18;
BО = AO = OC = 13 cm;
AO = OS = 13 cm, ⇒ AOC – isosceles;
Consider ONS – a right-angled triangle;
Let’s use the Pythagorean theorem:
HC² + OH² = OС²;
НС² = 169 – 25;
НС² = 144;
HC = 12 cm;
AC = An + НС;
AC = 12 + 12;
AC = 24 cm (since OH is the median);
S = 1/2 * AC * BH;
S = 1/2 * 24 * 18;
S = 12 * 18;
S = 216 cm;
Answer: S = 216 cm.