Problem. To cut off from a given circle (ABC) a segment which shall contain an angle equal to a given angle (X).
Sol.—Take any point A in the circumference. Draw the tangent AD, and make the angle DAC equal to the given angle X. AC will cut off the required segment.
Dem.—Take any point B in the alternate segment. Join BA, BC. Then the angle DAC is equal to ABC [xxxii.]; but DAC is equal to X (const.). Therefore the angle ABC is equal to X.