Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/157

Rh PROPOSITION 16. PROBLEM.

To inscribe an equilateral and equiangular quindecagon in a given circle.

Let ABCD be the given circle: it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD.

Let AC be the side of an equilateral triangle inscribed in the circle; [IV. 2. and let AB be the side of an equilateral and equiangular pentagon inscribed in the circle. [IV. 11.

Then, of such equal parts as the whole circumference ABCDEF contains fifteen, the arc ABC, which is the third part of the whole, contains five, and the arc AB, which is tht; fifth part of the whole, contains three; therefore their difference, the arc BC, contains two of the same parts. Bisect the arc BC at E; [III. 30. therefore each of the arcs BE, EC is the fifteenth part of the whole circumference ABCDF.

Therefore if the straight lines BE, EC be drawn, and straight lines equal to them be placed round in the whole circle, [IV. 1. an equilateral and equiangular quindecagon will be inscribed in it.

And, in the same manner as was done for the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon will be described about it; and also, as for the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it.