2. Prove that gu || Ac; and similarly Ac || FK, and GF or uk || BD, 3. that :. figs. GK, GC, AK, FB, BK are oms, 4. that each of gh, FK = GF or hk; and :: fig. gk is equilateral, 5. that _ AGB is a right 2, 6. that fig. gk is rectangular, 7. that .. GK is a square, and described about O ABCD. 1. Prove that the opposite sides of figs. AK, KE, AH, HD, AG, GC, BG, GD = each other, that AE = AF, and :. FG = GE, 3. that GE, GF, GH, GK = each other, that if a o be described with centre g, and either of these as distance, the sides AB, BC, CD, DA will touch the o, and ; the o is inscribed in the given square. PROPOSITION IX. Problem. To describe a circle about a given square. Steps of the Demonstration. 1. Prove that in AS ABC, ADC) / DAC = Z BAC; i. e., that , DAB is bisected by ac, 2. that similarly ZS ABC, BCD, and cda, are bisected by BD and ac, 3. that _ EAB = LEBA, and :. EA = EB, 4. that EA, EB, EC and Ed = each other; and :. a o described from centre E and distance either of these lines will pass through the extremities of the other three, and be described about the given square. PROPOSITION X. E Problem *. To describe an isosceles triangle, having each of the angles at the base double of the third angle. B * This is considered the most useful problem in Euclid. Steps of the Demonstration. 1. Prove that AB X BC = BD, 2. that BD touches the O ACD, 3. that / BDC = Z DAC, 4. that whole 2 BDA = L CDA + Z DAC, 5. that _ BDA = Z BCD, 6. that S BDA, DBA, and BCD = each other, 7. that BD = DC, and :. CA = DC, 8. that / CDA + DAC = 2 DAC, 9. that BCD = 2 / DAC, 10. that each of ZS BDA, DBA = 2 _ DAB. Steps of the Demonstration. 1. Prove that 5 L DAC, ACE, ECD, CDB, and BDA = each other, that AB, BC, CD, DE, EA = each other, 3. that the pentagon ABCDE is equilateral, 4. that ABD = EBD, 5. that / BAE = Z AED, 6. that the pentagon is equiangular, and is inscribed in the given circle. that, similarly, { and 2 CLD = 2 2 CLF, 1. Prove that each of s at c is a right L, 2. that, similarly, each of sat B and d are rt. ZS, 3. that Fcs + ck* = FB? + BK”, 4. that cK = BK, 5. that in AS FBK, FCK, \ Z BFK = KFC, land / BKF = _ FKC, 6. that :. _ BFC = 2 X KFC, and < + BKC = 2 / FKC, = 2 ., 7. 8. that _ BFC = L CFD, 9. that _ KFC = L CFL, 10. that in AS FKC, FLC) Kc = cl, and _ FKC = L FLC, 11. that kl = 2 Kc; and, similarly, HK = 2 BK, 12. that HK = KL, 13. that the pentagon is equilateral, 1 14. that / HKL = Z KLM, 15. that the pentagon is equiangular, and is described about the given O. Steps of the Demonstration. (base BF = base FD, 1. Prove that, in AS BCF, DCF, {and CBF = _ CDF, 2. that % CBA = 2 Z CBF, 3. that _ ABC is bisected by BF, 4. that, similarly, ZS BAE, AED, are bisected by AF, Fe, respectively, 5. that (in As FHC, FKC,) FH = FK, 6. that the five right lines, FG, FH, FK, FL, FM, = each other, and :: a o described from F, with either of these as distance, will pass through the extremities of the other four, and touch the sides AB, BC, CD, DE, EA. PROPOSITION XIV. B E Problem. To describe a circle about a given equilateral and equiangular pentagon. |