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Let P1 P2 P3 ... P10 be a regular polygon inscribed in a circle with radius 1. Compute

(P1 P2)^2 + (P2 P3)^2 + ... + (P1 P10)^2.

 

The sum includes all terms of the form (Pi Pi + 1)^2, where 1 <= i <= 9.  We write Pi Pj to mean the length of segment Pi Pj.)

 Mar 10, 2024

Best Answer 

 #1
avatar+399 
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Because P1P2P3P10is a regular polygon, all the side lengths are equal.

Therefore we only need to calculate (¯P1P2)210.

Suppose the center of the circle is O, then P1OP2=36.

Apply law of cosines:

(¯P1P2)2=12+1225+14=25+12.

(¯P1P2)210=2010(5+12)=20555=1555.

So our answer is 1555.

 Mar 10, 2024
 #1
avatar+399 
+3
Best Answer

Because P1P2P3P10is a regular polygon, all the side lengths are equal.

Therefore we only need to calculate (¯P1P2)210.

Suppose the center of the circle is O, then P1OP2=36.

Apply law of cosines:

(¯P1P2)2=12+1225+14=25+12.

(¯P1P2)210=2010(5+12)=20555=1555.

So our answer is 1555.

hairyberry Mar 10, 2024

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