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?

#Geometry

Easy Math Editor

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Comments

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 - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$