tircle, and DB touches the fame: The rectangle AD, DC is Book III. equal to the fquare of DB. D a 18. 3. =100 Either DCA paffes through the centre, or it does not; first, let it pass through the centre 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 rect angle AD, DC, together with the fquare of EC, is equal to the square of ED, and CE is equal to EB: There fore 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 ÈB, BD, becaufe 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 D b 6. 2. C 47. I. e f 3.3. But if DCA does not pafs through the centre of the circle ABC, take the centre E, and draw EF perpendicular to d 1. 3. AC, and join EB, EC, ED: And because the ftraight line EF, € 12. I. which paffes through the centre, cuts the ftraight line AC, which does not pass through the centre, at right angles, it fhall likewife bifectf it; there fore 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 thefe equals add the fquare of FE; there- B fore 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, A FE, becaufe EFD is a right angle; and the fquare of EC is equal to the fquares of CF, FE; therefore the rectangle AD, F DC, together with the fquare of EC, is equal to the fquare of G 2 ED: € 47. I. A of Book III. ED: But the fquares of EB, BD are equal to the fquare 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 F 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 straight 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. Draw the ftraight line DE touching the circle ABC, find its centre 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 square of and the straight line DE equal to the ftraight line DB: ᎠᏴ ; And Ꭰ d 8. I. and FE is equal to FB, wherefore DE, EF are equal to DB, Book III. BF; and the bafe 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. E c 16. 3. F A воок IV. 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 gular points of the figure about which it is defcribed, each III. A rectilineal figure is faid to be infcribed IV. A rectilineal figure is faid to be defcribed about a circle, when о VI. A circle is faid to be described about a rectilineal figure, when the circumference of the circle paffes through all the angular points of the figure about which it is defcribed. VII. A ftraight line is faid to be placed in a circle, when the extremities of it are in the circumference of the circle. Book IV. IN a given circle to place a straight line, equal to a given straight line not greater than the diameter of the circle, Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle. Draw BC the diameter of the circle ABC; then, if BC is equal to D, the thing required is done; for in the circle ABC a ftraight line BC is placed equal to D: But, if it is not, BC is greater than D; make CE equal to D, and from the centre C, at the distance CE, defcribe the circle AEF, and join CA: Therefore, becaufe C is the centre of the circle AEF, CA is equal to CE; but D is A E a 3. I. F equal to CE; therefore D is equal to CA: Wherefore, in the circle ABC, a ftraight line is placed equal to the given straight line D, which is not greater than the diameter of the circle. Which was to be done. IN a given circle to inscribe a triangle equiangular to a given triangle. |