Book III.

a 17.

b 23. I.

C 32.

It is required to cut off a fegment from the given circle ABC, that fhall contain an angle equal to the given angle D.

Draw the line EF, touching the circle in B; from which draw BC, making an angle FBC equal to the angle Db; then the angle FBC will be equal to the angle in the alternate feg. ment, viz. BAC; but FBC is equal to the angle D; therefore BAC is equal to the angle D. Wherefore, &c.

PROP. XXXV. THEOR.

IF
F two right lines in a circle mutually cut each other, the rec
tangle contained under the fegment of the one, is equal to the
rectangle contained under the fegments of the other.

Let the two right lines AC, DB, in the circle ABCD, mutually cut each other in E; then the rectangle under AE, EC, is equal to the rectangle under DE, EB; if AC, DB, pafs each through the center, then the rectangle under AE, ÉC, is equal to the rectangle under DE, EB, for the lines are ea def. 15. 1. qual.

b 3.

€ 47. J. d 5. 2.

2dly, If AC, paffing through the center, cut BD, not paffing through the center, at right angles, in the point E, find the center F, and join FD; for, because BE is equal to ED 6, and the angle DEF is a right one, the fquares of DE, EF, are equal to the fquare of FD ; but the rectangle under AE, EC, together with the fquare of EF, is equal to the fquare of FC, or FD. Take the fquare of FE, which is common, from both, there remains the rectangle under AE, EC, equal to the fquare of ED, that is, the rectangle under BE, ED. If the right line, AC, paffing through the center, cut BD, not pafling through the center, and not at right angles, draw FG at right angles to BD, and join FD; then BG is equal to GD; the rectangle under BE, ED, together with the fquare of GE, is equal to the fquare of GD, Add the fquare of GF to both, then the rectangle under BE, ED, with the fquares of EG, GF, or the fquare of LF are equal to the fquare of FD; the rectangle under AE, EC, together with the fquare of EF, are likewife equal to the fquare of D. Take the fquare of EF from both, then the rectangle under AE, EC, is equal to the rectangle under BE, ED.

but

3dly, If neither pass through the center, draw GH, paffing through the center F, and cutting AC, BD, in E; then the rectangle under AE, EC, is equal to the rectangle under BE, ED; for each is equal to the rectangle under GE, EH. Wherefore, &c.

PRO P. XXXVI. T HEOR.

IF fome point be taken without a circle, and from that point two right lines be drawn, one of which touches the circle, and the other cuts it, the rectangle under the whole fecant line, and the part between the point and convexity of the circle, is equal to the Square of the tangent line.

Let ABC be the circle, D the given point, and DCA, DB, the two given right lines, of which DB touches the circle, and DCA cuts it; the rectangle under AP, DC, is equal to the fquare of DB.

Book III.

Now, DCA either paffes thro' the center, or not. First, let it pass thro' the center E, and join BE; then, because AC is bifected in E, and DC added, the rectangle under AD, DC, together with the fquare of CE, are equal to the fquare of DE 2; a 6. but the fquare of DE is equal to the fquares of DB, BE; for b 47. I. the angle DBE is a right angle; therefore the rectangle under AD, DC, together with the fquare of CE, are equal to the fquares of DB, BE. Take the equal fquares of BE, CE, from both, there remains the rectangle AD, DC, equal to the square of DB.

C 18.

dr.

2dly, Let DA not pass through the center of the circle ABC; find the center Ed, and join ED, EC, EB; draw EF, at right € 12. Ú angles, to AC, cutting it in Fe; then AF is equal to FCF; 13. therefore the rectangle under AD, DC, together with the fquare of CF, are equal to the fquare of FD. Add the fquare of FE to both; then the rectangle under AD, DC, with the fquares of CF, FE, are equal to the fquares of DF, FE, but the fquare of CE is equal to the fquares of CF, FE, and the fquare of DE equal to the fquares of DF, FEb; therefore the rectangle under AD, DC, with the fquare of CE, are equal to the square of DE; but e fquare of DE is equal to the fquares of DB, BE; therefore the rectangle under AD, DC, with the fquare of CE, are equal to the fquares of DB, BE. Take the equal fquares of BE, CE, from both, and the rectangle under AD, DC, is equal to the fquare of DB. Wherefore, &c.

PROP

BOOK III.

a 17:

b 1. C 18.

d 36.
e Hyp.

£8. 1.

16.

[F, from a point without a circle two right lines be drawn, one of which cuts the circle, and the other falls upon it; and, if the rectangle under the whole fecant line, and part betwixt the point and circle, be equal to the fquare of the other line, this laft line Shall be a tangent to the circle.

Let fome point D, be affumed without the circle ABC, and from it draw the lines DCA, DB, so that DCA cut the circle, and DB fall upon it; and, if the rectangle under AD, DC, be equal to the fquare of DB, then DB will touch the circle in the point B.

For, let DE be drawn a tangent to the circle in the point E'; find the center Fb, and join BF, FE, and DF.

Then the angle DEF is a right angle ; therefore the rectangle under AD, DC, is equal to the fquare of DE 4; but the rectangle under AD, DC, is equal to the fquare of DB; therefore the fquare of DB is equal to the fquare of DE, and DB equal to DE; therefore the right lines DE, EF, are equal to DB, BF; and FD common; therefore the angle DBF is equal to the angle DEF f; but DEF is a right angle; therefore DBF is likewife a right angle: Therefore DB is a tangent to the circles. Wherefore, &c.

COR. I. Hence, if any number of right lines, as DA, DG, be drawn from the point A, cutting the circle in C and H, the rectangles under AD, DC, and GD, DH, are equal to one another; for each of them is equal to the fquare of BD.

II. If, from any two points in the circumference of a circle, two tangents be drawn, fo that, being produced, they will meet one another; then these tangents will be equal to one another; for each of their fquares, viz. of BD, DE, is equal to the rectangle contained under AD, DC.

1

THE