2. Prove that GH || AC; and similarly AC || FK, and 3. 4. 5. 6. 7. GF or HK || BD, that. figs. GK, GC, AK, FB, BK are □ms, that each of GH, FK = GF or нк; and .. fig. GK is equilateral, that AGB is a right 4, that fig. GK is rectangular, that :. GK is a square, and described 1. Prove that the opposite sides of figs. AK, KE, AH, 2. 3. 4. Hd, ag, gc, bg, GD = each other, that AE AF, and .. FG = GE, that GE, GF, GH, GK = each other, that if a 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. i. e., that 1. Prove that (in AS ABC, ADC) DAC = BAC; DAB is bisected by ac, that similarly s ABC, BCD, and CDA, are bisected by BD and ac, 2. 3. 4. that EAB ▲ EBA, and .. EA = EB, that EA, EB, EC and ED each other; and .. a 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 This is considered the most useful problem in Euclid. Steps of the Demonstration. 1. Prove that AB X BC = BD2, that BD touches the O ACD, S BDA, DBA, and BCD = each other, 2. 3. that BDC 4. that whole that BD 10. CDA + ≤ dac = 2 ≤ dac, BCD 2 ≤ DAC, that each of LS BDA, DBA = 2 ≤ dab. PROPOSITION XI. H B E Problem. To inscribe an equilateral and equiangular pentagon in a given circle. Steps of the Demonstration. 1. Prove that 5 ≤S DAC, ACE, ECD, CDB, and BDA = each other, 2. 3. 4. 5. 6. that AB, BC, CD, DE, EA = each other, that the pentagon ABCDE is equilateral, that the pentagon is equiangular, and is inscribed in the given circle. 2. 3. Steps of the Demonstration. 1. Prove that each of s at c is a right, that, similarly, each of ▲ s at B and D are rt. ≤s, that FC + CK = FB2 + BK, 7. 8. 10. 11. 12. 13. 14. 15. CFD2CFL, that, similarly, and CLD = 22 CLF, that (in AS FKC, FLC) KC = CL, and ▲ FKC = FLC, that KL = 2 KC; and, similarly, HK = 2 BK, that HK KL, that the pentagon is equilateral, j that HKL = / KLM, that the pentagon is equiangular, and is described about the given . (base BF base FD, CBF = CDF, 1. Prove that, in As BCF, DCF, and that ▲ CBA = 2 ≤ cbf, ABC is bisected by BF, that, similarly, S BAE, AED, are bisected by AF, FE, respectively, 2. 3. that 4. 5. 6. that (in As FHC, FKC,) FH = FK, EA. PROPOSITION XIV. Problem. To describe a circle B about a given equilateral and E F equiangular pentagon. |