Book II. tained by BE, EF is equal to the square of EH: but the rectangle contained by BE, EF is BD, for EF is equal to ED: therefore the parallelogram BD is equal to the fquare of HE; but BD is equal to the rectilineal figure A: therefore the rectilineal figure A is equal to the fquare defcribed upon EH. Wherefore a square is made equal to the given rectilineal figure A, the square described upon EH. Which was to be done. I. DEFINITION S. EQUAL circles, are those of which the diameters are Book III. are equal. equal; or of which the ftraight lines from the centers 2. A ftraight line is faid to touch a circle, which meeting the circle, and being produced does not cut the circle. 3. Circles are faid to touch one another, which, meeting each other, do not cut one another. 4. In a circle straight lines are said to be equally distant from the center, when the perpendiculars drawn from the center upon them are equal. 5. But that line is faid to be more distant upon which the greater perpendicular falls. 6. A segment of a circle is the figure bounded by a straight line and the circumference of a circle. 7. But an angle of a segment is that contained by a straight line and the circumference of a circle. VOL. I. 8. But I Book III. 8. But an angle in a fegment is; when any point is taken in the circumference of the fegment, and from it ftraight lines are joined to the extremities of the straight line which is the base of the segment; the angle contained by the straight lines fo joined. 9. But when the straight lines containing the angle receive any circumference, the angle is faid to ftand upon that. 10. And a fector of a circle is, when an angle ftands at the center of the circle, the figure bounded by the ftraight lines containing the angle, and the circumference intercepted by them. 11. Similar fegments of a circle are fuch as receive equal angles; or in which the angles are equal to one another. Let the given circle be the circle ABC: it is required to find the center of the circle. Let any straight line AB be drawn in it as it may happen; and let it be cut in halves at the point D, and from the point D let DC be drawn at right angles to AB, and let it be produced to E; and let CE be cut in halves at the point F: I fay that F is the center of the circle ABC. C FG For if not, but if poffible let it be G; and let GA, GD, GB be joined; and becaufe AD is (by conft.) equal to DB, and DG common; certainly the two AD, DG are equal to the two GD, DB each to each; and the base GA is (by def. 15.1.) equal to the base GB; for they are from the center G; wherefore the angle ADG is equal to the angle GDB; but when a ftraight line ftanding npon a straight line makes the adjacent angles equal to one another, each of the equal angles is a right angle (by def. 10. 1.); therefore GDB is a right angle; but FDB is alfo a right angle; wherefore FDB is equal to GDB, the greater to the lefs, which is impoffible; therefore the point G is not the center A D B E of of the circle ABC: certainly in the fame manner we shall demon- Book III. strate that neither is any other but the point F. Wherefore the point F is the center of the circle. Which was to be done. Cor. Certainly from this it is manifeft, that if in a circle any ftraight line, cut any straight line in halves and at right angles, the center of the circle is in the cutting line. PRO P. II. If two accidental points be taken in the circumference of a circle; the ftraight line joining these points will fall within the circle. Let ABC be a circle, and let two accidental points be taken in the circumference of it, as the points A, B; I say that the straight line, joining A and B, will fall within the circle. For if not, but if poffible let AEB fall without; and let the center of the circle ABC be taken (by 1. 3.), and let it be D; and let AD, DB be joined; and let DF be produced to E. And fince DA is equal to DB, therefore the angle DAE (by 5. 1.) is equal to DBE; and because one fide of the triangle DAE is viz produced, the fide AEB, therefore (by 16. E D C B F 1.) the angle DEB is greater than DAE: Wherefore if two accidental points be taken in the circum- Book III. PRO P. III. If in a circle any straight line through the center cut in halves any straight line not through the center, it will also cut it at right angles; and if it cut it at right angles it will alfo cut it in halves. Let ABC be a circle, and let CD any straight line in it through the center cut in halves AB any ftraight line, not through the center, in the point F; I say that it alfo cuts it at right angles. For let the center of the circle ABC be taken (by 1.3.); and let it be E, and let EA, EB be joined. And because AF is equal to FB, and FE C E F B D halves AB not paffing through the center, also cuts it at right angles. But let CD cut AB at right angles; I say that it also cuts it in halves; that is, that AF is equal to FB. For the same things being constructed, because EA from the center is equal to EB, the angle alfo EAF is equal (by 5. 1.) to EBF; but the right angle AFE is equal to the right angle BFE; therefore there are two triangles EAF, EBF having the two angles equal to the two angles, and one fide equal to one fide, viz. EF common to them, extended under one of the equal angles; they will therefore alfo have (by 26. 1.) the remaining fides equal to the remaining fides; wherefore AF is equal to BF. Wherefore if in a circle, any ftraight line through the center cut in halves any straight line not through the center; it will also cut it at right angles; and if it cut it at right angles it will also cut it in halves. Which was to be demonftrated. PROP. |