Plane trigonometry, by S.L. Loney.

[Exs. XXXVIII.] REGULAR POLYGONS.

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12. If a, b, c, d be the sides of a quadrilateral, taken in order, prove
that
d2= a2 + b2 + c' - 2ab cos a - 2bc cos f - 2ca cos y,
where a, p and y denote the angles between the sides a and b, b and c,
and c and a respectively.
223. Regular Polygons. A regular polygon is a
polygon which has all its sides equal and all its angles
equal.
If the polygon have n angles we have, by Euc. I. 32,
Cor., n times its angle + 4 right angles = twice as many
right angles as the figure has sides = 2n right angles.
2n-4 2n-4 ' r
Hence each angle = right angles = x -
n ~ 2
radians.
224. Radii of the inscribed and circumscribing circles
of a regular polygon,
Let AB, BC, and CD be three successive sides of the
polygon, and let n be the A D
number of its sides.
Bisect the angles ABC \ /
and BCD by the lines BO R/ i R./
and CO which meet in 0,
and draw OL perpendicular B i
to BC. L
It is easily seen that 0 is the centre of both the
incircle and the circumcircle of the polygon and that
BL equals LC.
Hence we have OB = OC = R, the radius of the circumcircle and OL = r, the radius of the incircle.