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 inscribed о in a rectilineal figure, when the circumference of the circle touches each side of the figure. VI. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described. VII. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle. 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, any 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 +Hyp. to D: but if it is not, BC is greater than D: make CE *3. 1. equal to D, and from the centre C, at the distance CE, describe the circle AEF, and join CA: D CA shall be equal to D. Because C is the centre of the circle AEF, CA is +15 Def 1. equal to CE: but D is equal to CE; therefore *Const. D is equal to CA. Wherefore in the circle ABC, +1 Ax. 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. 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. Draw the straight line GAH, touching the *17.3. circle in the point A, and at the point A, in the straight line AH, make* *23. 1 the angle HAC equal to the point A, in the straight line quired. H B 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 #32.3. +1 Ax. *Const. 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 *32. 1. BAC is equal to the remaining angle EDF: wherefore the triangle ABC is equiangular to the triangle DEF, and it is inscribed in the circle ABC. Which was to be done. and 1 Ax. 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. +1. 3. #23. 1. #17.3. Produce EF both ways to the points G, H; findt 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, 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; these lines will meet and form a triangle LMN, which 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: consequently the lines AM, BM will make with a line joining A, B, angles which are together less than two right angles, these straight lines must therefore meet.* In like manner, AL, CL meet, as also BN, CN and because the four angles of the quadrilateral figure AMBK are equal to four right angles #18. 3. 112 Ax. for it can be divided into two triangles; and that two of them, KAM, KBM are right angles; therefore the other I D +3 Ax. #13. 1. +1 Ax. +Const. +3 Ax. two AKB, AMB are equal to two right angles : but the angles DEG, DEF are likewise equal to two right angles; therefore the angles AKB, AMB are equal to the angles DEG, DEF; of which AKB is equal† to DEG; wherefore the remaining angle AMB is equal† to the remaining angle DEF. In like manner, the angle LNM may be demonstrated to be equal to DFE; 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. #32. 1. and 3 Ax. 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. #9. 1. #12 Ax. Bisect the angles ABC, BCA by the straight lines BD, CD meeting one another in the point D, from which draw a perpendicular DE to one of the sides. Then the circle described with centre D and distance DE will be inscribed in the triangle. For #12. 1. draw DF, DG perpendiculars to the other sides. 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 +11 Ax. the right angle BFD; therefore E 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, B which is opposite to one of the equal #26. 1. +1 Ax. angles in each, is common to both; therefore their other sides are equal;* wherefore DE is equal to DF: for a like 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 it will 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: #16.3. therefore the straight lines AB, BC, CA do each of them touch the circle: they also form a triangle because every two meet, since they form with a line joining the points of contact, angles on the same side together less than two right angles, 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. #10. 1. 11. 1. Bisect AB, AC in the points D, E, and from these points draw DF, EF at right angles to AB, AC. Then DF, EF must meet one another, because they make, with a line joining D, E, angles on the same side, M |