IV. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal. V. And the straight line on which the greater perpendicular falls, is said to be farther from the centre. VI. A segment of a circle is the figure con tained by a straight line and the circumference it cuts off. VII. "The angle of a segment is that which is contained by "the straight line and the circumference." VIII. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment. IX. And an angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle. X. A sector of a circle is the figure contained by two straight lines drawn from the centre, and the circumference between them. XI. Similar segments of circles are those in which the angles are equal, or which contain equal angles. PROP. I. PROB. To find the centre of a given circle. Let ABC be the given circle; it is required to find its centre. Draw any chord AB, and bisect it in D; from the * 10. 1. point D draw* DC at right angles to AB, and produce * 11. 1. CD to E, and bisect CE in F: the point F shall be the centre of the circle ABC. For, if it be not, let, if possible, G be the centre, and join GA, GD, GB: then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG are equal to the two BD, DG, each to each; and the base GA is F D equal to the base GB, because they B D E 8. 1. are drawn from the centre G: there- A fore the angle ADG is equal to the angle GDB: but when a straight line, standing upon another straight line, makes the adjacent angles equal to one another, each of the angles is a right angle: *10 Def. therefore the angle GDB is a right angle: but FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impossible: therefore G is not the centre of the circle ABC. In the same manner it can be shown, Whenever the expression "straight lines from the centre," or "drawn from the centre," occurs, it is to be understood that they are drawn to the circumference. 1. that no other point, which is not in the line CE, is the centre; the centre is therefore in CE, and any other point than F divides CE into unequal parts, and cannot be the centre; therefore F is the centre of the circle ABC. Which was to be found. COR. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other.. 1. 3. * 5. 1. PROP. II. THEOR. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. с Let ABC be a circle, and A, B any two points in the circumference; the straight line drawn from A to B shall fall within the circle. A F E B For, if it do not, let it fall, if possible, without, as AEB; find * D the centre of the circle ABC, and join DA, DB. In the circumference AB take any point F; join DF, and produce it to meet AB in E: then, because DA is equal to DB, the angle DAB is equal to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater than the angle DAE; but DAE was proved equal to the angle DBE: therefore the angle DEB is greater than the angle DBE; but the greater * 19. 1. angle is subtended by the greater side;* therefore DB is greater than DE: but DB is equal to DF; wherefore DF is greater than DE, the less than the greater, which is impossible: therefore the straight line drawn from A to B does not fall without the circle. In the same manner, it may be demonstrated that it does not 16. 1. that no other point, which is not in the line CE, is the centre; the centre is therefore in CE, and any other point than F divides CE into unequal parts, and cannot be the centre; therefore F is the centre of the circle ABC. Which was to be found. COR. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other. * 1. 3. 5. 1. PROP. II. THEOR. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference; the straight line drawn from A to B shall fall within the circle. * * A F E B For, if it do not, let it fall, if possible, without, as AEB; find* D the centre of the circle ABC, and join DA, DB. In the circumference AB take any point F; join DF, and produce it to meet AB in E: then, because DA is equal to DB, the angle DAB is equal to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater than the angle DAE; but DAE was proved equal to the angle DBE: therefore the angle DEB is greater than the angle DBE; but the greater 19. 1. angle is subtended by the greater side ;* therefore DB is greater than DE: but DB is equal to DF; wherefore DF is greater than DE, the less than the greater, which is impossible: therefore the straight line drawn from A to B does not fall without the circle. In the same manner, it may be demonstrated that it does not 16. 1. |