Sidebilder
PDF
ePub

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

[blocks in formation]

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

[blocks in formation]

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,

[blocks in formation]

S BDA, DBA, and BCD = each other,

2.

3.

that BDC

4.

that whole

[blocks in formation]

that BD

[blocks in formation]

10.

[blocks in formation]

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,

[blocks in formation]

that the pentagon is equiangular, and is inscribed in the given circle.

[blocks in formation]

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,

[blocks in formation]

7.

8.

[blocks in formation]
[blocks in formation]

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 .

[blocks in formation]

(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,
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.

Problem. To describe a circle B about a given equilateral and

E

F

equiangular pentagon.

« ForrigeFortsett »