In equal circles, angles, whether at the centre or circumference, which stand upon equal arches, are equal. F E K PROP. XXVIII. THEOR. In equal circles, equal chords cut off equal PROP. XXX. PROB. To bisect a given arch of a circle. PROP. XXXI. THEOR. The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle, is an acute angle; and the angle in a segment less than a semicircle, is an obtuse angle. B PROP. XXXII. THEOR. If a straight line touches a circle, and from the point of contact a straight line is drawn, cutting the circle, the angles made by this cutting line with the tangent are equal to the angles in the alternate segments of the circle. E PROP. XXXIII. PROB. Upon a given straight line to describe a segment of a circle containing an angle equal to a given angle. F B To cut off from a given circle a segment which shall contain a given angle. B PROP. XXXV. THEOR. If two chords in a circle intersect each other, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. PROP. XXXVI, THEOR. If from a point outside of a circle two straight lines be drawn, the one cutting the circle, the other touching it, the rectangle contained by the whole secant or cutting line and its external segment is equal to the square of the tangent. If from a point outside of a circle two straight lines be drawn, the one cutting the circle, the other meeting it, and if the rectangle contained by the whole cutting line and its external segment be equal to the square of the line meeting the circle, the latter is a tangent to the circle. F G B |