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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 P1OP2=36010=36.

 

Then P1P22=12+122(1)(1)cos36=21+52=352. Note that (512)2=352. Then P1P2=512.

Repeating the same process 10 times gives P1P2+P2P3+P3P4++P9P10+P10P1=10512=5(51).

 May 2, 2024

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