Again, since the arc AF is equal to ED, to each add the arc ABCD; then the whole arc FABCD is equal to ABCDE; therefore the angles DEF, EFA which stand on these arcs are equal [III. xxvii.]. In the same manner it may be shown that the other angles of the hexagon are equal. Hence it is equiangular, and is therefore a regular hexagon inscribed in the circle.
Cor. 1.—The side of a regular hexagon inscribed in a circle is equal to the radius.
Cor. 2.—If three alternate angles of a hexagon be joined, they form an inscribed equilateral triangle.
Exercises.
1. The area of a regular hexagon inscribed in a circle is equal to twice the area of an equilateral triangle inscribed in the circle; and the square of the side of the triangle is three times the square of the side of the hexagon.
2. If the diameter of a circle be produced to C until the produced part is equal to the radius, the two tangents from C and their chord of contact form an equilateral triangle.
3. The area of a regular hexagon inscribed in a circle is half the area of an equilateral triangle, and three-fourths of the area of a regular hexagon circumscribed to the circle.
PROP. XVI.—Problem.
To inscribe a regular polygon of fifteen sides in a given circle.
Sol.—Inscribe a regular pentagon ABCDE in the circle [xi.], and also an equilateral triangle AGH [ii.]. Join CG. CG is a side of the required polygon.
