A circle is inscribed in an -sided regular polygon having each side unit

MATHEMATICS · Subjective · ISC · Mathematics · Sets and Functions · Compound and multiple angles

Problem

A circle is inscribed in an nn-sided regular polygon A1,A2,…,AnA_1, A_2, \dots, A_n having each side aa unit. For any arbitrary point PP on the circle, prove that ∑i=1n(PAi)2=na24(1+cos⁡2(π/n)sin⁡2(π/n)).\sum_{i=1}^{n} (PA_i)^2 = n \frac{a^2}{4} \left( \frac{1 + \cos^2(\pi/n)}{\sin^2(\pi/n)} \right).

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