From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.
Proof
Let the circle and angle θ be given. Draw a tangent BC
to the circle by III.17. At the point of contact B, construct
∠CBD=θ in the half-plane that intersects the circle
(I.23). Let BD meet the circle at D. The chord BD cuts off
two segments; the segment on the far side of BD from the tangent
admits the inscribed angle θ by III.32 (tangent-chord
angle).