Angle ABC inscribed in a circle, O-center of the circle. Chord AB = m and ∠ACB = α / 2. Find the radius of the circle.

Since the inscribed angle ACB, by condition, is equal to α / 2, and it rests on the arc AB, then the central angle AOB is equal to half the degree measure of the angle ACB.

Angle AOB = α0.

The AOB triangle is isosceles, since ОА = ОВ = R.

Determine the radius of the circle by the cosine theorem for a triangle.

AB ^ 2 = ОА ^ 2 + ОВ ^ 2 – 2 * ОА * ОВ * CosАВ.

m ^ 2 = 2 * R ^ 2 – 2 * R ^ 2 * Cos α.

m ^ 2 = 2 * R ^ 2 * (1 – Cos α).

R ^ 2 = m ^ 2 / (2 * (1 – Cos α)).

R = m / √ (2 * (1 – Cos α)).

Answer: The radius of the circle is m / √ (2 * (1 – Cos α)).