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Steps of the Demonstration.

1. Prove that FA, FB, FE, bisect ZS CBA, BAE, AED,

2.

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3.

4.

F

E.

that FCD= / CDF,

that. FC FD,

that the five right lines, FA, FB, FC, FD, fe

each other; and .. a O described from centre F and distance any one of them, will pass through the points A, B, C, D, E and bedescribed about the given pentagon.

B

PROPOSITION XV.

Problem. To inscribe an equilateral and equiangular hexagon in a given circle.

Steps of the Demonstration.

1. Prove that A EGD is equilateral,

that

EGD

of 2 rights, and similarly

DGC = of 2 rights,

that CGB = of 2 rights,

2.

3.

4.

that the

and. S EGD, DGC, CGB = each other,

vertical to these are to them,

and the six s = each other,

that.. AB, BC, CD, DE, EF, FA, = each other,

that the hexagon is equilateral,

5.

6.

7.

that FBD ECA, and .. AFE

8.

FED,

that the hexagon is equiangular; and it is inscribed in the given circle.

B

E

PROPOSITION XVI.

Problem. To describe an equilateral and equiangular quindecagon in a given circle.

Steps of the Demonstration*.

1. Prove that if the be divided into 15 equal parts,

2.

3.

ABC will contain 5 such parts, and AB will contain 3 such parts, and.. their difference BC will contain 2 such parts.

that. BC being bisected, BE or EC will contain one such part.

that if right lines BE Or EC be drawn, and their equals extended round the whole ; an equilateral and equiangular pentagon will be inscribed in it.

It should be observed that the triangle and pentagon must be so placed in the circle, that one angle of each may meet in the same point A.

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