A triangle is inscribed in a circle of radius R, the vertices of which divide the circle in a ratio of 2: 5: 17.

A triangle is inscribed in a circle of radius R, the vertices of which divide the circle in a ratio of 2: 5: 17. Find the area of a triangle

Let the triangle ABC be inscribed in a circle, and the vertices A, B and C divide the circle into parts 2: 5: 17. Let the angle C rest on the arc AB is 2 parts of a circle with a center O and a radius R.
Determine the angles A, B, C. Arc AB = 360 * 2 / (2 + 5 + 17) = 30 (degrees). <C = 1/2 (arcs AB) = 30/2 = 15.
Arc AC = 17 * 360/24 = 17 * 15 = 255, <B = 255/2 = 127.5
Arc BC = 5 * 360/24 = 150, <A = 150/2 = 75
AB / sin (<C = 15) = BC / sin (<A = 37.5) = AC / sin (<B = 127.5) = 2R.
BC = BC = R * 2 * sin37.5 = 0.52R,
AC = AC = R * 2 * sin127.5 = 1.587R
The area of the triangle ABC = BC * AC * sin (<C) / 2 = 1/2 * AC x BC x sin15 = 1/2 * 1.587 * R * 1.2178 * R * 0.2589 = 0.251 * R ^ 2.