tilineal figure, when the circumference of the circle touches each side of the figure. VI. A circle is said to be described about a rec- о VII. + Hyp. 3. 1. + Constr. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle. PROP. I. PROB. In a given circle to place a straight line, equal to a given straight line which is 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: it is required to place in the circle ABC a straight line equal to D. 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 is greater than D: make CE equal to D, and from the centre C, at the distance CE, describe the circle AEF, and join CA: CA shall be equal to D. * B Because C is the centre of the circle AEF, CA is +15 Def. 1. equal + to CE: but D is equal† to CE; therefore D is equal to CA. Wherefore in the circle ABC, a straight line CA is placed equal to the given straight line D, which is not greater than the diameter of the circle. Which was to be done. +1 Ax. PROP. II. PROB. In a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF. * H * 23. 1. Draw the straight line GAH touching the circle in 17.3. the point A, and at the point A, in the straight line AH, make* the angle HAC equal to the angle DEF; and at the point A, in the straight line AG, make the angle E GAB equal to the angle DFE; and join BC: ABC shall be the triangle required. 32. 3. Constr. +1 Ax. Because HAG touches the circle ABC, and AC is drawn from the point of contact, the angle HAC is equal to the angle ABC in the alternate segment of the circle: but HAC is equal + to the angle DEF; therefore also the angle ABC is equal + to DEF: for the same reason, the angle ACB is equal to the angle DFE: therefore the remaining angle BAC is equal to 32. 1. & the remaining angle EDF: wherefore the triangle ABC 1 Ax. is equiangular to the triangle DEF, and it is inscribed in the circle ABC. Which was to be done. PROP. III. PROB. * About a given circle to describe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to describe a triangle about the circle ABC equiangular to the triangle DEF. Produce EF both ways to the points G, H; find † †1. 3. the centre K of the circle ABC, and from it draw any straight line KB; at the point K in the straight line KB, make* the angle BKA equal to the angle DEG, *23.1. and the angle BKC equal to the angle DFH; and through the points A, B, C, draw the straight lines LAM, MBN, NCL, touching* the circle ABC: LMN 17.3. shall be the triangle required. Because LM, MN, NL touch the circle ABC in the points A, B, C, to which from the centre are drawn KA, KB, KC, the angles at the points A, B, C, are right angles: and because the four angles of the drilateral figure AMBK are equal to four right angles, for it can be divided into two triangles; and that two qua 18. 3. +3 Ax. * 13.1. + 1 Ax. + Constr. + 3 Ax. 3 Ax. of them KAM, KBM are right angles, therefore the GE M B N to the remaining angle DEF. In like manner, the angle LNM may be demonstrated to be equal to DFE; *32. 1. & and therefore the remaining angle MLN is equal to the remaining angle EDF: therefore the triangle LMN is equiangular to the triangle DEF: and it is described about the circle ABC. Which was to be done. See N. *9. 1. * 12.1. † 11 Ax. *26. 1. † 1 Ax. 16, 3. PROP. IV. PROB. To inscribe a circle in a given triangle. Let the given triangle be ABC; it is required to inscribe a circle in ABC. E B Bisect * the angles ABC, BCA by the straight lines BD, CD meeting one another in the point D, from which draw* DE, DF, DG perpendiculars to AB, BC, CA. And because the angle EBD is equal to the angle FBD, for the angle ABC is bisected by BD, and that the right angle BED is equal+ to the right angle BFD; therefore the two triangles EBD, FBD have two angles of the one equal to two angles of the other, each to each; and the side BD, which is opposite to one of the equal angles in each, is common to both; therefore their other sides are equal*; wherefore DE is equal to DF: for the same reason, DG is equal to DF: therefore DE is equal to DG: therefore the three straight lines DE, DF, DG, are equal to one another, and the circle described from the centre D, at the distance of any of them, will pass through the extremities of the other two, and touch the straight lines AB, BC, CA, because the angles at the points E, F, G, are right angles, and the straight line which is drawn from the extremity of a diameter at right angles to it, touches the circle: therefore the straight lines AB, * BC, CA do each of them touch the circle, and therefore the circle EFG is inscribed in the triangle ABC. Which was to be done. PROP. V. PROB. To describe a circle about a given triangle. Let the given triangle be ABC; it is required to describe a circle about ABC. See N. Bisect* AB, AC, in the points D, E, and from these *10. 1. points draw DF, EF at right angles to AB, AC; DF, *11. 1. EF, produced meet one another: for, if they do not meet, they are parallel, wherefore AB, AC, which are at right angles to them, are parallel; which is ab * C B surd: let them meet in F, and join FA; also if the point F be not in BC, join BF, CF. Then, because AD is equal to DB, and DF common, and at right angles to AB, the base AF is equal to the base FB. In *4. 1. like manner, it may be shewn that CF is equal to FA; and therefore BF is equal † to FC; and FA, FB, FC, †1Ax. are equal to one another: wherefore the circle described from the centre F, at the distance of one of them, will pass through the extremities of the other two, and be described about the triangle ABC. Which was to be done. COR. And it is manifest, that when the centre of the circle falls within the triangle, each of its angles is less than a right anglet, each of them being in a segment † 31. 3. greater than a semicircle; but, when the centre is in one, of the sides of the triangle, the angle opposite to this side, being in a semicirclet, is a right angle; and, if the centre falls without the triangle, the angle opposite to the side beyond which it is, being in a segment less than a semicirclet, is greater than a right angle: therefore, conversely, if the given triangle be acute angled, the centre of the circle falls within it; if it be a right angled triangle, the centre is in the side opposite to the right angle; and if it be an obtuse angled triangle, the centre falls without the triangle, beyond the side opposite to the obtuse angle, To inscribe a square in a given circle. Let ABCD be the given circle; it is required to inscribe a square in ABCD. Draw the diameters, AC, BD†, at right angles to one another, and join AB, BC, CD, DA: the figure ABCD shall be the square required. Because BE is equal to ED, for E is the centre, and that EA is common, and at right angles to BD; the base BA is equal * to the base AD: and for the same reason, BC, CD are each of them equal to BA, or AD; therefore the quadrilateral figure ABCD is equilateral. It is also rectangular; for the straight line BD being the diameter of the circle ABCD, BAD is a semicircle; wherefore the angle BAD is a right* angle: for the same reason, each of the angles ABC, BCD, CDA, is a right angle: therefore the quadrilateral figure B ABCD is rectangular: and it has been shewn to be equilateral; therefore it is at square; and it is inscribed in the circle ABCD. Which was to be done. PROP. VII. PROB. E To describe a square about a given circle. Let ABCD be the given circle; it is required to describe a square about it. Draw two diameters AC, BD of the circle ABCD, at right angles to one another, and through the points A, B, C, D, draw FG, GH, HK,KF touching the circle: the figure GHKF shall be the square required. * Ꮐ Because FG touches the circle ABCD, and EA is drawn from the centre E to the point of contact A, the angles at A are right angles: for the same reason, the angles at the points B, C, D are right angles: and because the angle AEB is a right angle, as likewise is EBG, GH is parallel* to AC: for the same reason AC is parallel to FK and in like manner GF, HK may B each of them be demonstrated to be parallel to BED: therefore the figures GK, GC, AK, FB, BK are pa |