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Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

 May 2, 2024
 #1
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Suppose O is the centre of the circle. Then ΔOP1P2 is an isosceles triangle with OP1=OP2=1 and .

 

Then . Note that . Then .

Repeating the same process 10 times gives .

 May 2, 2024

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