and because BK is equal to KC, as was demonstrated, and that KL is double of KC, and HK double of BK, (Ax. 6.) 16. therefore 17. GH, GM, ML, are each of them equal to HK or KL: The pentagon GHKLM is equilateral. It is also equiangular; for, since the angle FKC is equal to the angle FLC, and the angle HKL double of the angle FKC, and KLM double of FLC, as was before demonstrated, (Ax. 6.) 1. The angle HKL is equal to KLM. And in like manner it may be shown, that 2. Each of the angles KHG, HGM, GML, is equal to the angle HKL or KLM; therefore, the five angles GHK, HKL, KLM, LMG, MGH, being equal to one another, 3. The pentagon GHKLM is equiangular : and it is equilateral, as was demonstrated; and it is described about the circle ABCDE. Q.E.F. PROP. XIII.-PROBLEM. To inscribe a circle in a given equilateral and equiangular pentagon. Let ABCDE be the given equilateral and equiangular pentagon; it is required to inscribe a circle in the pentagon ABCDĚ. Bisect (I. 9.) the angles BCD, CDE, by the straight lines CF, DF, and from the point F, in which they meet, draw the straight lines FB, FA, FE. A Therefore, since BC is equal (Hyp.) to CD, and CF common to the triangles BCF, DCF, the two sides BC, CF, are equal to the two DC, CF, each to each; and the angle BCF is equal (Constr.) to the angle DCF; therefore (I. 4.) 1. The base BF is equal to the base FD, and the other angles to the other angles, to which the equal sides are opposite; therefore 2. The angle CBF is equal to the angle CDF: and because the angle CDE is double of CDF, and that CDE is equal to CBA, and CDF to CBF; also therefore wherefore 3. CBA is double of the angle CBF; 4. The angle ABF is equal to the angle CBF; 5. The angle ABC is bisected by the straight line BF. In the same manner it may be demonstrated that 6. The angles BAE, AED, are bisected by the straight lines AF, FE. From the point F draw (I. 12.) FG, FH, FK, FL, FM, perpendiculars to the straight lines AB, BC, CD, DE, EA; and because the angle HCF is equal to KCF, and the right angle FHC equal to the right angle FKC; in the triangles FHC, FKC, there are two angles of one equal to two angles of the other, each to each; and the side FC, which is opposite to one of the equal angles in each, is common to both; therefore the other sides shall (I. 26.) be equal, each to each; wherefore 7. The perpendicular FH is equal to the perpendicular FK. In the same manner it may be demonstrated that therefore 8. 9. FL, FM, FG, are each of them equal to FH or FK: The five straight lines FG, FH, FK, FL, FM, are equal to one another; wherefore the circle described from the centre F, at the distance of one of these five, shall pass through the extremities of the other four, and touch the straight lines AB, BC, CD, DE, EA, because the angles at the points G, H, K, L, M, are right angles, and that a straight line drawn from the extremity of the diameter of a circle at right angles to it, touches (III. 16.) the circle; therefore 10. Each of the straight lines AB, BC, CD, DE, EA, touches the circle; wherefore it is inscribed in the pentagon ABCDE. Q.E.F. PROP. XIV.-PROBLEM. To describe a circle about a given equilateral and equiangular pentagon. Let ABCDE be the given equilateral and equiangular pentagon; it is required to describe a circle about it. Bisect (I. 9.) the angles BCD, CDE, by the straight lines CF, FD, and from the point F, in which they meet, draw the straight lines FB, FA, FE, to the points B, A, E. It may be demonstrated, in the same manner as in the preceding proposition, that 1. The angles CBA, BAE, AED, are bisected by the straight lines FB, FA, FE; and because the angle BCD is equal to the angle CDE, and that FCD is the half of the angle BCD, and CDF the half of CDE; ́(Ax. 7.) 2. The angle FCD is equal to FDC; wherefore (I. 6.) 3. The side CF is equal to the side FD. In like manner it may be demonstrated that therefore 4. FB, FA, FE, are each of them equal to FC or FD; 5. The five straight lines FA, FB, FC, FD, FE, are equal to one another; and the circle described from the centre F, at the distance of one of them, shall pass through the extremities of the other four, and be described about the equilateral and equiangular pentagon ABCDE. PROP. XV.-PROBLEM. Q.E.F. To inscribe an equilateral and equiangular hexagon in a given circle. Let ABCDEF be the given circle; it is required to inscribe an equilateral and equiangular hexagon in it. Find (I. 3.) the centre G of the circle ABCDEF, and draw the diameter AGD; and from D as a centre, at the distance DG, describe the circle EGCH; join EG, CG, and produce them to the points B, F; and join AB, BC, CD, DE, EF, FA; the hexagon ABCDEF is equilateral and equiangular. Because G is the centre of the circle ABCDEF, 1. GE is equal to GD; and because D is the centre of the circle EGCH, 2. DE is equal to DG; wherefore (Ax. 1.) 3. GE is equal to ED, and the triangle EGD is equilateral; and therefore (I. 5. Cor.) 4. The three angles EGD, GDE, DEG, are equal to one another, but the three angles of a triangle are equal (I. 32.) to two right angles : therefore 5. The angle EGD is the third part of two right angles; in the same manner it may be demonstrated that also 6. The angle DGC is the third part of two right angles : and because the straight line GC makes with EB the adjacent angles EGC, CGB, equal (I. 13.) to two right angles, 7. The remaining angle CGB is the third part of two right angles; therefore 8. The angles EGD, DGC, CGB, are equal to one another; and (I. 15.) 9. To EGD, DGC, CGB, are equal the vertical opposite angles BGA, AGF, FGE; therefore 10. The six angles EGD, DGC, CGB, BGA, AGF, FGE, are equal to one another: but equal angles stand (III. 26.) upon equal circumferences; therefore 11. The six circumferences AB, BC, CD, DE, EF, FA, are equal to one another; and equal circumferences are subtended (III. 29.) by equal straight lines ; therefore the six straight lines are equal to one another, and 12. The hexagon ABCDEF is equilateral. It is also equiangular; for, since the circumference AF is equal to ED, to each of these add the circumference ABCD; therefore 1. The whole circumference FABCD is equal to the whole EDCBA: and the angle FED stands upon the circumference FABCD, and the angle AFE upon EDCBA; therefore (III. 27.) 2. The angle AFE is equal to FED: in the same manner it may be demonstrated that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED; therefore 3. The hexagon ABCDEF is equiangular; and it is equilateral, as was shown; and it is inscribed in the given circle ABCDEF. Q.E.F. COR. From this it is manifest, that the side of the hexagon is equal to the straight line from the centre, that is, to the semidiameter of the circle. And if through the points A, B, C, D, E, F, there be drawn straight lines touching the circle, an equilateral and equiangular hexagon shall be described about the circle, which may be demonstrated from what has been said of the pentagon; and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon. PROP. XVI.-PROBLEM. To inscribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD. B Let AC be the side of an equilateral triangle inscribed (IV. 2.) in the circle, and AB the side of an equilateral and equiangular pentagon inscribed (IV. 11.) in the same; therefore 1. Of such equal parts as the whole circumference ABCDF contains fifteen, the circumference ABC contains five, it being the third part of the whole, and 2. The circumference AB contains three parts, it being the fifth part of the whole; therefore 3. Their difference BC contains two of the same parts. Bisect (III. 30.) BC in E; therefore 4. BE, EC, are, each of them, the fifteenth part of the whole circumference ABCD; therefore if the straight lines BE, EC, be drawn, and straight lines equal to them be placed around (IV. 1.) in the whole circle, an equilateral and equiangular quindecagon shall be inscribed in it. Q.E.F. And in the same manner as was done in the pentagon, if, through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about the circle; and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it. |