Question

A,B,C,D,E,F are 6 consecutive points on the circumference of a circle such that AB=BC=CD=10,DE=EF=FA=22. If the radius of the circle is √n, what is the value of n?

Details and assumptions:
-The lengths given are side lengths, not arcs.

Answer 1

Let the center of the circle be O.

Let 2θ = ∠AOB
Let 2φ = ∠EOF
6θ+6φ = 2π
θ+φ = π/3
sin(θ+φ) = √3/2

Now,
in ∆AOB, sinθ = 5/r
in ∆EOF, sinφ = 11/r

sin(θ+φ) = sinθcosφ + cosθsinφ
so

5/r * √(r^2-121)/r + 11/r * √(r^2-25)/r = √3/2
5√(r^2-121) + 11√(r^2-25) = √3/2 r^2
r^2 = 268