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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
1 year 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. Calvin Lin Staff · 1 year ago

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