Book III. dicular to AC; therefore AG is equal to GC; wherefore w the rectangle AE, EC, together with the fquare of EG, is equal to the fquare of AG: To each of thefe equals add the fquare of GF; therefore the rectangle AE, EC, together with the fquares of EG, GF is equal to a 3.3. b 5.2. € 47. I. A E C G B the fquares of AG, GF: But the D H F D been fhewn, to the rectangle GE, rectangle BE, ED is equal to the EH; and, for the fame reafon, the A E C G B fame rectangle GE, EH; therefore the rectangle BE, ED. Wherefore, if two ftraight lines, &c. Q. E. D. IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, fhall be equal to the fquare of the line which touches it. Let D be any point without the circle ABC, and DCA, DB two ftraight lines drawn from it, of which DCA cuts the circle, circle, and DB touches the fame: The rectangle AD, DC is Book III. equal to the fquare of DB. D Either DCA paffes through the center, or it does not; first, let it pass through the center E, and join EB; therefore the angle EBD is a right angle: And because the ftraight line AC is bifected in E, and produced to the point D, the rectangle AD, DC, together with the fquare of EC, is equal to the square of ED, and CE is equal to EB: Therefore the rectangle AD, DC, together B with the fquare of EB, is equal to the fquare of ED: But the fquare of ED, is equal to the fquares of EB, BD, becaute EBD is a right angle: Therefore the rectangle AD, DC, together with the fquare of EB, is equal to the fquares of EB, BD: Take away the common fquare of EB; therefore the remaining rectangle AD, DC is equal to the fquare of the tangent DB. a 18. 3. b 6.2. E C 47. I. A f 3. 3. But if DCA does not pafs through the center of the circle ABC, take the center E, and draw EF perpendicular to d 1.3. AC, and join EB, EC, ED: And because the ftraight line EF, e 12. 1. which paffes through the center, cuts the straight line AC, which does not pass through the center, at right angles, it fhall likewife bifect fit; therefore AF is equal to FC: And because the ftraight line AC is bifected in F, and produced to D, the rectangle AD, DC, together with the fquare of FC, is equal to the fquare of FD: To each of these equals add the fquare of FE; therefore the rectangle AD, DC, together with the fquares of CF, FE, is equal to the fquares of DF, FE: But the fquare of ED is equal to the fquares of DF, FE, because EFD is a right angle; and the fquare of EC is equal to the fquares of CF, FE; therefore the c F E rectangle AD, DC, together with the fquare of EC, is equal to the fquare of ED: And CE is equal to EB; therefore the rectangle AD, DE, together with the fquare of LB, is equal G 2 to € 47. I. A Book III. to the fquare of ED: But the fquares of EB, BD are equal to the fquare of ED, because EBD is a right angle; therefore the rectangle AD, DC, together with the fquare of EB, is equal to the fquares of EB, BD: Take away the common fquare of EB; therefore the remaining rectangle AD, DC is equal to the fquare of DB. Wherefore, if from any point, &c. Q. E. D. COR. If from any point without a circle, there be drawn two ftraight lines cutting it, as AB, AC, the rectangles contained by the whole lines and the parts of them without the circle, are equal to one another, viz. the rectangle BA, AE to the rectangle CA, AF: For each of them is equal to the fquare of the straight line AD which touches the circle. D E F C B a 17. 3. b 18. 3. € 36. 3. IF from a point without a circle there be drawn two ftraight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle be equal to the fquare of the line which meets it, the line which meets fhall touch the circle. Let any point D be taken without the circle ABC, and from it let two ftraight lines DCA and DB be drawn, of which DCA cuts the circle, and DB meets it; if the rectangle AD, DC be equal to the fquare of DB; DB touches the circle. a Draw the ftraight line DE touching the circle ABC, find its center F, and join FE, FB, FD; then FED is a right angle: And because DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal to the fquare of DE: But the rectangle AD, DC is, by hypothefis, equal to the fquare of DB: Therefore the fquare of DE is equal to the fquare of DB, and the ftraight line DE equal to the straight line DB: And D d 8. I. and FE is equal to FB, wherefore DE, EF are equal to DB, Book III. BF; and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF is equal to the angle DBF; but DEF is a right angle, therefore alfo DBF is a right angle: And FB, if produced, is a diameter, and the ftraight line which is drawn at right angles to a diameter, from the B extremity of it touches the circle: Therefore DB touches the circle ABC. Wherefore, if from a point, &c. Q.E. D. F E e 16. A DEFINITION S. I. Rectilineal figure is faid to be infcribed in another rectilineal figure, when all the angles of the infcribed figure are upon the fides of the figure in which it is infcribed, each upon each. II. In like manner, a figure is faid to be defcribed III. A rectilineal figure is faid to be inscribed IV. A rectilineal figure is faid to be defcribed about a circle, when each fide of the circumfcribed figure touches the circumference of the circle. V. In like manner, a circle is faid to be infcri- |
