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.)
Because P1P2P3…P10is a regular polygon, all the side lengths are equal.
Therefore we only need to calculate (¯P1P2)2∗10.
Suppose the center of the circle is O, then ∠P1OP2=36∘.
Apply law of cosines:
(¯P1P2)2=12+12−2∗√5+14=2−√5+12.
(¯P1P2)2∗10=20−10(√5+12)=20−5√5−5=15−5√5.
So our answer is 15−5√5.
Because P1P2P3…P10is a regular polygon, all the side lengths are equal.
Therefore we only need to calculate (¯P1P2)2∗10.
Suppose the center of the circle is O, then ∠P1OP2=36∘.
Apply law of cosines:
(¯P1P2)2=12+12−2∗√5+14=2−√5+12.
(¯P1P2)2∗10=20−10(√5+12)=20−5√5−5=15−5√5.
So our answer is 15−5√5.