A Treatise On Plane And Spherical Trigonometry And Its Applications to Astronom

Edward a Edward Albert Bowser

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DE MOIVRWS PROPERTY OF THE CIRCLE. 247 170. De Moivre's Property of the Circle. Let be the centre of a circle, P any point in its plane. Divide the circumference into D n equal parts EC, CD, DE, , begin- ning at any point B; and join and P with the points of division B, G, D, .... Let POB = 6; then will OB 2M -2 OB" OP" cos ?i0+OP 2n = PB 2 PC 2 PD 2 . To n terms.
For, put OB = a, OP = a;, and = -; then PB 2 = OP 2 + OB 2 - 2 OP . OB cos = x- + a 2 2 ax cos - n (1) PC 2 = OP 2 + OO 2 - 2 OP OC ...cos = x 2 -f a 2 2 ax cos ; and so on n Multiplying (1), (2), (3), " together, we have PB 2 PC 2 PD 2 to n terms = [or 2 ax cos - + n ax cos - 2 ax cos + a 2 = a 2n - 2 a"^ cos + a 2 " . [by (4) of Art. 169] B 2n ... (3) which proves the proposition.
248 PLANE TRIGONOMETRY.
171. Cote's Properties of the Circle. These are particu- lar cases of De Moivre's property of c C the circle.
(1) Let OP, produced if necessary, meet the circle at A, and let n then nO is a multiple of 2?r. Hence we have from (3) of Art.

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