Assume that $A (i = 1, 2, \dots, n)$ be the vertices of a regular polygon inscribed in a unit circle. Find the value of ${|A_{1} A_{2}|}^{2} + {|A_{1} A_{3}|}^{2} + \dots + {|A_{1} A_{n}|}^{2}$ .

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Detailed Solution

Assume that the circle centred at the origin

$O A_{1} = e^{i \frac{2 π}{n}}, O A_{2} = e^{i \frac{4 π}{n}}$

Thus,

$\begin{array}{l} |\vec{A_{1} A_{p}}| = |\vec{O A_{p}} - \vec{O A_{1}}| \\ = |e^{i \frac{2 p π}{n}} - e^{i \frac{2 π}{n}}| \\ = |e^{i \frac{2 π}{n}} (e^{i (2 p - 1) \frac{π}{n}} - 1)| \\ = |(e^{i (2 p - 1) \frac{π}{n}} - 1)| \\ = |2 i \sin \{(p - 1) \frac{π}{n}\} e^{i (p - 1) \frac{π}{n}}| \end{array}$

$= 2 \sin ((p - 1) \frac{π}{n})$

Therefore

${|A_{1} A_{2}|}^{2} + {|A_{1} A_{3}|}^{2} + {|A_{1} A_{4}|}^{2} + \dots + {|A_{1} A_{n}|}^{2} \begin{array}{l} = \sum_{p = 1}^{n} 4 \sin^{2} (p - 1) \frac{π}{n} \\ = \sum_{p = 1}^{n} 2 (1 - \cos \{2 (p - 1) \frac{π}{n}\}) \\ = \sum_{p = 2}^{n} 2 (1 - \cos \{2 (p - 1) \frac{π}{n}\}) \end{array}$

$\begin{array}{l} = 2 [(n - 1) - \frac{\sin ((n - 1) \frac{π}{n})}{\sin (\frac{π}{n})} \times \cos (\frac{2 (\frac{2 π}{n}) + 2 (n - 2) \frac{π}{n}}{2}]] \\ = 2 [(n - 1) - \frac{\sin (π - \frac{π}{n})}{\sin (\frac{π}{n})} \times \cos (π)] \end{array}$