a 18. 3. --/ circle, and DB touches the fame : The rectangle AD; DC is Book 111. equal to the square of DB. Either DCA passes through the centre, or it does not ; first, Jet it pass through the centre E, and join EB; therefore the angle EBD is a right angle: And because D the straight line AC is bifected in E, and produced to the point D, the recta 22 angle AD,. DC, together with the b 6. 2. square 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 square of EB, is equal to the {quare of ED: But the square of ED E is equal to the squares of EB, BD, be C 47. I. cause 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 А common square of EB; therefore the remaining rectangle AD, DC is equal to the square of the tangent DB. 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 EF, • 12. 1. which passes through the centre, cuts the straignt line AC, which does not pass through the centre, at right D angles, it shall likewise bisedtf it; there. f 3. 3• fore 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 F with the squares of CF, FE, is equal to E the squares of DF, FE: But the quare of ED is equal to the squares of DF, A FE, becaule EFD is a right angle; and the square of EC is equal 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 of ED: G 2 1 Book III. ED : But the squares of EB, BD are equal to the square of WED, because EBD is a right angle; therefore the rectangle C 47. I. 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 А E С B PRO P. XXXVII. THEOR. IF 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 shall touch the circle. 2 17. 3 b 18. 3 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 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 becaule DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal c to the square of DE: But the rečiangle AD, DC is, by hypothelis, 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 line DB: And € 36. 3. and FE is equal to FB, wherefore DE, EF are equal to DB, Book IT. BF; and the base FD is common to the two criangles DEF, DBF; there D fore the angle DEF is equal d to the d 8. I. 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 right angles to a diameter, from the B E e 16. 3 extremity of it, touches the circle: Therefore DB toucbes the circle ABC. Wherefore, if from a point, &c. R. E. D. F Ε L Ε Μ Ε Ν . Τ N T S 1. Rectilineal figure is said to be inscribed in another reati lineal figure, when all the angles of the inscribed figure II. about another figure, when all the Gdes of III. in a circle, when all the angles of the in- IV. each side of the circumscribed figure V. In like manner, a circle is said to be infcri. bed in a rectilineal figure, when the circumference of the circle touches each Gde of the figure. 103 Book IV. VI. tilineal figure, when the circumference VII. mities of it are in the circumference of the circle. IN a given a given circle to place a straight line, equal to a gi ven itraight 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 straight line BC is placed equal to D: But, if it is not, BC A is greater than D; make CE equal' to D, and from the centre C, at the distance CE, de E a 3. I. scribe the circle AEF, and join B CA: Therefore, because Ċ is F the centre of the circle AEF, CA is equal to CE; but D is D equal to CE; therefore D is equal to CA: Wherefore, in the circle ABC, a straight 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. |