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Coroll. From hence it is manifeft, that the side of the Hexagon is equal to the Semidiameter of the Circle.

And if we draw thro' the Points A, B, C, D, E, F, Tangents to the Circle, an equilateral and equiangular Hexagon will be described about the Circle, as is manifest from what has been said concerning the Pentagon. And so likewise may a Circle bei infcribed and circumscribed about a given Hexagon ;

which was to be done. PROPOSITION XVI.

PROBLEM. To describe' an equilateral and cquiangular Quin

decagon in a given Circle. ET ABCD be a Circle given. It is tequired

to describe an equilateral and equiangular Quindecagon in the fame.

Let AC be the side of an equilateral Triangle inscribed in the Circle ABCD, and A B the Side of a Pentagon. Now if the whole Circumference of the Circle ABCD be divided into fifteen equal Parts, the Circumference ABC, one third of the whole, shall be five of the said fifteen equal Parts; and the Circumference AB, one fifth of the whole will be three of the faid Parts. Wherefore the remaining Circumference B C, will be two of the said Parts. And if BC be bifected in the Point E, BE, or EC, will be one fifteenth Part of the whole Circumference ABCD. And so if BE, EC, be joined, and either EC, or EB, be continually applied in the Circle, there Thall be an equilateral and equiangular Quindecagon de fcribed in the Circle ABCD, which was to be done.

If, according to what has been said of the Pentagon,

Right Lines are drawn thro’ the Divisions of the Circle touching the same, there will be described about the Circle an equilateral and equiangular Quindecagon. And, moreover, a Circle may be inscribed, or circumscribed, about a given equilatecal and equiangular Quindecagone

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