Book II. Then the rectangle under BE, EF, together with the w square of GE, are equal to the square of GFC, or GHd, but the square of GH is equal to the squares of GE, EH °; d Def. 15 therefore the rectangle under BE, EF, together with the {quare of GC, are equal to the squares of HE, EG. Take the square of GE from both, and the rectangle under BE, EF, that is, BD, is equal to the square of EH. Where. fore, &c. 1. THE EQUAL 1. Book III. III. IV. center, when perpendiculars drawn from the center to each V. Definition 19th, 1. 'Vİ. An angle of a segment is the angle contained by the right line and circumference of the circle. VII. An angle is said to be in a segment, when right lines are drawn from some point in the circumference to the ends of that line which is the base of the segment, which lines contain the angle. VIIT. Buty Book III. VIII. part of the circumference, then the angle is said to stand upon IX. right lines, drawn from the center, and the circumference be- X. or whereof the angles in them are equal. PRO P. I, PRO B. To find the centre of a circle. a 10. I. bal. I. Required to find the centre of the given circle ABC. Draw in it the line AB, which bisect in D“, by the right line CD at right angles to ABS, and produce CD to E; bifect EC in F; which point is the centre of the circle ABC. If not, let G be the centre ; join GB, GA, GD; then, be cause AD is equal to DB, DG is common, the base AG is ec def. 15. 1. qual to GB, and the angle ADG to GDBd; therefore, each of them is a right angle; but FDB is a right angle; therefore, GDB is equal to FDB, a part to the whole, which is impofe Ax. 9. 1. fible ; therefore, no point but F can be the centre. Where fore, &c. Cor. Hence, if, in a circle, any right line cut another right line into two equal parts, the centre of the circle will be in that line which cuts the other into two equal parts. d 8.1. PRO P. II. THE O R. If any, two points be allumed in the circumference of a circle , the right line joining these points will fall within the circle. Let the circle be ABC, A and B the points in its circumference, the right line AB, joining these points, will fall within the circle. Find D the centre of the circle join DA, DB, and draw DF, cutting the right line AB in the point E; then b def. 15. 1. the right lines DA, DB, DF, are equall. But DF is greater c AX. 9. 1. than DEC; therefore, DA, DB, are likewise greater than DE; but DB, DA, reach the circumference ; therefore DE does not |