a 3. 3. b 5. 2. c 47. 1. D A a F E G B Book III. pendicular to AC: therefore AG is equal to GC; wherefore the rectangle AE, EC, together with the square of EG, is equal to the square of AG: To each of these equals add the square of GF; therefore the rectangle AE, EC, together with the squares of EG, GF, is equal to the squares of AG, GF: But the squares of EG, GF are equal to the square of EF; and the squares of AG, GF are equal to the square of AF: Therefore the rectangle AE, EC, together with the square of EF, is equal to the square of AF; that is, to the square of FB: But the square of FB is equal to the rectangle BE, ED, together with the square of EF; therefore the rectangle AE, EC, together with the square of EF, is equal to the rectangle BE, ED, together with the square of EF: Take away the common square of EF, and the remaining rectangle, AE, EC, is therefore equal to the remaining rectangle BE, ED. H Lastly, Let neither of the straight lines AC, BD pass through the centre: Take the centre F, and through E, the intersection of the straight lines AC, DB, draw the diameter GEFH: And because the rectangle AE, EC is equal, as has been shown, to the rectangle GE, EH: and, for the same reason, the rectangle BE, ED is equal to the same rectangle GE, EH; therefore the rectangle AE, EC is equal to the rectangle F D E C G B BE, ED. Wherefore, " if two straight lines," &c. Q. E. D. PROP. XXXVI. THEOR. 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, is equal to the square of the line which touches it. Let D be any point without the circle ABC, and DCA, DB two straight lines drawn from it, of which DCA cuts the circle, and DB touches the same: The rectangle AD, DC is Book III. equal to the square of DB. Either DCA passes 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 straight line AC is bi sected in E, and produced to the с b 6. 2. E c 47. 1. A Take away the common square of EB: therefore the remaining rectangle AD, DC is equal to the square of the tangent DB. e D f 3. 3. But if DCA does not pass 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 straight line e 12. 1. EF, which passes through the centre, cuts the straight line AC which does not pass through the centre at right angles, it likewise bisects it; therefore AF is equal to FC: And because the straight line AC is bisected in F, and produced to D, the rectangle AD, DC, together with the square of FC, is equal to the square of FD: To each of these equals, add the square of FE; there- B fore the rectangle AD, DC, together with the squares of CF, FE, is equal to the squares of DF, FE: But the square of ED is equal to the squares of DF, FE, because EFD is a right angle: and the square of EC is equal F E to the squares of CF, FE; therefore the rectangle AD, DC, together with the square of EC, is equal to the square of ED: And CE is equal to EB; therefore the rectangle AD, DC, together with the square of EB, is equal to the square c 47. l. Book III. of ED: But the squares of EB, BD are equal to the square of ED, because EBD is a right angle; therefore the rectangle AD, DC, together with the square of EB, is equal to the squares of EB, BD: Take away the common square of EB; therefore the remaining rectangle AD, DC is equal to the square of DB. Wherefore, "if from any point," &c. Q. E. D. COR. If from any point without a circle, there be drawn two straight 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 these rectangles is equal to the square of the straight line AD which touches the circle. A E F D C B See N. a 17. 3. b 18. 3. c 36. 3. PROP. XXXVII. THEOR. If from a point without a circle there be drawn two straight 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 square of the line which meets it, the line which meets, also touches 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 square of DB; DB touches the circle. Draw a the straight 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 square of DE: But the rectangle AD, DC is, by hypothesis, equal to the square of DB: Therefore the square of DE is equal to the square of DB; and the straight line DE equal to the straight D d 8. 1. line DB. And FE is equal to FB, wherefore, DE, EF are Book III. equal to DB, BF; and the base FB 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 also DBF is a right angle: And FB, if produced, is a diameter, and the straight line which is drawn at B right angles to a diameter, from the extremity of it, touches the circle: Therefore DB touches the circle ABC. Wherefore, "if from a point," &c. Q. E. D. E e Cor. 16.3. F THE ELEMENTS OF EUCLID. Book IV. See N. A BOOK IV. DEFINITIONS. I. RECTILINEAL figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon each. II. In like manner, a figure is said to be described III. A rectilineal figure is said to be inscribed IV. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle. V. In like manner, a circle is said to be in- о |