1. Prove that the opposite sides of figs. AK, KP, AH,

HD, AG, GC, BG, GD = each other, that AE = AF, and .. FG = GE, 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,

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PROPOSITION IX.

Problem. To describe a circle about a given square.

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Steps of the Demonstration. 1. Prove that (in As ABC, ADC) – DAC = { BAC;

i. e., that 2 DAB is bisected by AC, that similarly S ABC, BCD, and cda, are

bisected by BD and ac, that EAB = Z EBA, and .. EA = EB, 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.

* This is considered the most useful problem in Euclid.

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Steps of the Demonstration. 1. Prove that 5 ZS DAC, ACE, ECD, CDB, and BDA

= each other, that AB, BC, CD, DE, EA = each other, that the pentagon ABCDE is equilateral, that ABD = EBD, that BAE = Z AED, that the pentagon is equiangular, and is

inscribed in the given circle.

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PROPOSITION XIII.

Problem. To inscribe a circle Le in a given equilateral and equi☺ angular pentagon.

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Steps of the Demonstration. 1. Prove that, in As BCF, DCF, {

base BF = base FD,

DCF, and < CBF = Z CDF, that CBA = 2 % CBF, that _ ABC is bisected by BF, that, similarly, LS BAE, AED, are bisected

by AF, Fe, respectively, that (in As FHC, FKC,) FH = FK, 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.

20 s.

PROPOSITION XIV.

Problem. To describe a circle Bk about a given equilateral and equiangular pentagon.