Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/156

132 But equal angles stand on equal arcs; [III. 26. therefore the six arcs AB, BC, CD, DE, EF, FA are equal to one another. And equal arcs are subtended by equal straight lines; [III.29. therefore the six straight lines are equal to one another, and the hexagon is equilateral.

It is also equiangular. For, the arc AF is equal to the arc ED; to each of these add the arc ABCD; therefore the whole arc FABCD is equal to the whole arc ABCDE; and the angle FED stands on the arc FABCD, and the angle FED stands on the arc ABCDE; therefore the angle FED is equal to the angle AFE. [III. 27.

In the same manner it may be shewn that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED; therefore the hexagon is equiangular. And it has been shewn to be equilateral; and it is inscribed in the circle ABCDEF.

Wherefore an equilateral and equiangular hexagon has been inscribed in the given circle,

. From this it is manifest that the side of the hexagon is equal to the straight line from the centre, that is, to the semidiameter of the circle.

Also, if through the points A, B, C, D, E, F, there be drawn straight lines touching the circle, an equilateral and equiangular hexagon will be described about the circle, as may be shewn from what was said of the pentagon; and a circle may be inscribed in a given equilateral and equir angular hexagon, and circumscribed about it, by a method like that used for the pentagon.