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# Regular Polygons and Incircles

Let $$P_1P_2 \cdots P_n$$ be a $$n$$-sided regular polygon (with a side length of say $$a$$) and X be an arbitrary point on its incircle.

(1) What would be $$\sum_{i=1}^n |XP_i|^2$$?, if $$|PQ|$$ denotes the length of the line segment $$\overline{PQ}$$.

Hint : It seems to be independent of the location of $$X$$!!

(2) Can the above conclusion be proven analytically?

Note by Janardhanan Sivaramakrishnan
2 years ago

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Great question. I think I know what your inspiration was.

Hint: In fact, one can show more generally that $$\sum |XP_i|^2$$ is dependent only on $$|X|$$ (assuming polygon is centered at the origin).

There is a nice proof using vectors.

Staff - 2 years ago