Geometrical Proof

Here is an interesting geometry problem-

Let $A_1$ , $A_2 \ \cdots \ A_n$ be the vertices of a regular $n$ -sided polygon $P$ . Let $X$ be a random point on the incircle of $P$ .

Prove that $\displaystyle\sum_{i=1}^{n} \overline{A_iX}^2=nR^2\left[1+\cos^2\left(\dfrac{\pi}{n}\right)\right]$ .

Note : Here, $R$ denotes the circumradius of $P$ .

#Geometry #Polygons #Circles

Easy Math Editor

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@KARAN SHEKHAWAT Here is the solution :

Let $O$ represent the center of the polygon.

Now, $\begin{aligned}\vec{XA_i}&=\vec{XO}+\vec{OA_i}\\ |\vec{XA_i}|^2&=|\vec{XO}|^2+|\vec{OA_i}|^2+2(\vec{XO} \cdot \vec{OA_i})\\ \sum_{i=1}^{n} |\vec{XA_i}|^2&=n(r^2+R^2)+2\sum_{i=1}^{n}(\vec{XO} \cdot \vec{OA_i})\\ &=n(r^2+R^2)+2\vec{XO} \cdot \left(\sum_{i=1}^{n} \vec{OA_i}\right)\end{aligned}$ Since $\displaystyle\sum_{i=1}^{n} \vec{OA_i}=0$ , $\sum_{i=1}^{n} |\vec{XA_i}|^2=\sum_{i=1}^{n} \overline{A_iX}^2=n(R^2+r^2)$

All that's left to do is to express $r$ in terms of $R$ . That's a matter of simple trigonometry.

Note 1 : $r$ is the inradius of $P$ .

Note 2 : $\displaystyle\sum_{i=1}^{n}(\vec{XO} \cdot \vec{OA_i})=\vec{XO} \cdot \left(\sum_{i=1}^{n} \vec{OA_i}\right)$ due to the distributivity of the dot product over vector addition.

Pratik Shastri - 5 years, 1 month ago

Awesome ! Thanks a lot ! It's is much better than complex number tecnique atleast for me :) You are genius !

Karan Shekhawat - 5 years, 1 month ago

Can we use particular case of polygon in which vertex are n'th roots of of unity To prove it ?

Karan Shekhawat - 5 years, 1 month ago

You could do that. We are to prove the general result which can also be, to my knowledge, proven that way.

Pratik Shastri - 5 years, 1 month ago

I'm get stuck in that ! If you Don't mind Can you please Upload its solution So that we can learn from it ?

Thanks !

Karan Shekhawat - 5 years, 1 month ago

@Karan Shekhawat – Well, I didn't solve it using complex numbers, I used vectors.

Pratik Shastri - 5 years, 1 month ago

@Karan Shekhawat – Anyways I will upload it within a day or two.

Pratik Shastri - 5 years, 1 month 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}`	$\boxed{123}$