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Steps of the Demonstration. 1. Prove that Fa, FB, FE, bisect S CBA, BAE, AED,

that _ FCD = 2 CDF,
that :. FC = FD,
that the five right lines, FA, FB, FC, FD, FE

= each other; and :. a 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.

PROPOSITION XV.

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

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o

Steps of the Demonstration. 1. Prove that A EGD is equilateral,

that / EGD = of 2 right <s, and similarly

< DGC = f of 2 right ZS, that < CGB = { of 2 right ZS,

and :: S EGD, DGC, CGB = each other, that the Zs 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, that FBD = Eca, and :: L AFE = _ FED, that the hexagon is equiangular; and it is

inscribed in the given circle.

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PROPOSITION XVI.

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

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Steps of the Demonstration *. 1. Prove that if the o be divided into 15 equal parts,

ABC will contain 5 such parts, and AB will contain 3 such parts, and :: their differ

ence Bc will contain 2 such parts. that :: BC being bisected, Be or EC will con

tain one such part. that :: if right lines BE or Ec be drawn, and

their equals extended round the whole O; an equilateral and equiangular pentagon will be inscribed in it.

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* 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.

THE END.

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