Book III. translations from it, the angle of the greater segment is faid to be greater, and the angle of the lefs fegment is faid to be less than ⚫ a right angle.' COR. From this it is manifest, that if one angle of a triangle be equal to the other two, it is a right angle, because the angle adjacent to it is equal to the fame two; and when the adjacent angles are equal, they are right angles. I a Fa ftraight line touches a circle, and from the point of II. I. angles made by this line with the line touching the circle, hall be equal to the angles which are in the alternate segments of the circle. Let the ftraight line EF touch the circle ABCD in B, and from the point B let the straight line BD be drawn cutting the circle. the angles which BD makes with the touching line EF shall be equal to the angles in the alternate fegments of the circle; that is, the angle FBD is equal to the angle which is in the fegment DAB, and the angle DBE to the angle in the fegment BCD. A D From the point B draw BA at right angles to EF, and take any point Cin the circumference BD, and join AD, DC, CB; and because the ftraight line EF touches the circle ABCD in the point B, and BA is drawn at right angles to the touching line from the point of contact B, the center of the circle is in BA; therefore the angle ADB in a femic. 31. 3. circle is a right angle, and confequently the other two angles BAD, b. 19.3. c 32. 1. ABD ́are equal to a right angle. 42 D the angles BAD, ABD. take from these equals the common angle DBE, being likewife equal f to two right angles, are equal to the Book III. angles BAD, BCD; and DBF has been proved equal to BAD; therefore the remaining angle DBE is equal to the angle BCD in the f. 13. 1, alternate fegment of the circle. Wherefore if a straight line, &c, Q. E. D. UPON PROP. XXXIII. PROB. 'ON a given straight line to describe a fegment of See N a circle, containing an angle equal to a given rectilineal angle. Let AB be the given straight line, and the angle at C the given rectilincal angle; it is required to describe upon the given straight line AB a fegment of a circle, containing an angle equal to the angle C. a First, Let the angle at C be a right angle, and bisect AB in F, and from the center F, at the distance FB defcribe the femicircle AHB; therefore the angle AHB in a femicircle is bequal to the right angle at C. C H But if the angle C be not a right angle, at the point A in the ftraight line AB make the angle BAD equal to the angle C, and C. 23. I. d. 11. I 0. 4. 1. to the base GB; and the circle described from the center G, at the distance GA fhall pafs thro the point B; let this be the circle AHB, and because from the point A the extremity of the diameter AE, AD is drawn at right angles to AE, therefore AD touches f the circle; f. Cor. 16.3. and because AB drawn from the point of contact A cuts the circle, which contains an angle equal to the given angle at C. Which was to be done. 17.3. то O cut off a segment from a given circle which fhall contain an angle equal to a given rectilineal angle. Let ABC be the given circle, and D the given rectilineal angle; it is required to cut off a fegment from the circle ABC that shall contain an angle equal to the angle D. a Draw the straight line EF touching the circle ABC in the point B, and at the point B, in the b. 23. 1. ftraight line BF, make the angle FBC equal to the angle D. therefore because the circle ABC, and BC is D contact B, the angle FBC is c t. 32. 3. equal to the angle in the alternate segment BAC of A E B F C the circle. but the angle FBC is equal to the angle D; therefore the angle in the fegment BAC is equal to the angle D. wherefore the fegment BAC is cut off from the given circle ABC containing an angle equal to the given angle D. which was to be done. PROP. PROP. XXXV. THEOR. Book III, IF two ftraight lines within a circle cut one another, the See N rectangle contained by the fegments of one of them, is equal to the rectangle contained by the fegments of the other. Let the two straight lines AC, BD within the circle ABCD, cut one another in the point E; the rectangle contained by AE, EC is equal to the rectangle contained by BE, ED. If AC, BD pafs each of them thro' the center, fo that E is the center; it is evident, A E C that AE, EC, BE, ED being all equal, the B rectangle AE, EC is likewife equal to the rectangle BE, ED. But let one of them BD pass thro' the center, and cut the other AC, which does not pafs thro' the center, at right angles, in the point E. then if BD be bisected in F, F is the center of the circle ABCD; and to it from A draw AF. and because BD which passes thro the center cuts the straight line AC which does not pass thro' of FA; but the fquares of AE, EF are B C. 47. I. equal to the fquare of FA; therefore the rectangle BE, ED together with the fquare of EF is equal to the squares of AE, EF. take away the common fquare of EF, and the remaining rectangle BE, ED is equal to the remaining fquare of AE; that is, to the rectangle AE, EC. Next, Let BD which paffes thro' the center, cut the other AC, which does not pass thro' the center, in E, but not at right angles. then, as before, if BD be bifected in F, F is the center of the circle. Join AF, and from F draw FG perpendicular to AC; there d. 12. 1. fore a Book III. fore AG is equal to GC; wherefore the rectangle AE, EC toge ther with the fquare of EG is equal b to the fquare of AG. to each of thefe equals add the fquare of GF, therefore the rectangle AE, EC together with the squares of EG, a. 3. 3. b. 5.2. Ꭰ to the square of FB. but the fquare of FB is equal to the rectangle fquare of EF, and the remaining rectangle AE, EC is therefore equal Lastly, Let neither of the straight lines AC, BD pafs thro' the center. take the center F, and thro' E the interfection of the ftraight lines H AC, DB draw the diameter GEFH. A D E C G B EC is equal to the rectangle BE, ED. Wherefore if two straight lines, &c. Q. E. D. PRO P. XXXVI. THEOR. F from any point without a circle two straight lines IF 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, and ᎠᏴ |