The distance (AiX^{2}) from point X to vertice Ai is igiven by the rule of cosine
AiX^{2} = R^{2} + r^{2}  2Rr cos(ai) where aiX is the angle XOAi, O is the centre of both circles, R is the circumradius and r is the inradius
A_{1}X^{2} + A_{2}X^{2} + ... + A_{n}X^{2} = nR^{2}+nr^{2 } 2Rr [cos(a1)+cos(a2)+ ...+cos(an)]
The term with the sum of cosines is always 0 (with a square the cosines are opposed by pairs, with a triangle for a different reason). So:
A_{1}X^{2} + A_{2}X^{2} + ... + A_{n}X^{2} = nR^{2}+nr^{2 }
But r=Rcos(pi/n). So:
A_{1}X^{2} + A_{2}X^{2} + ... + A_{n}X^{2} = nR^{2}(1 + cos^{2}(pi/n))
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Edited on April 7, 2019, 4:59 pm

Posted by armando
on 20190407 16:57:57 