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 \(P\).
Note by
Pratik Shastri
3 years, 7 months ago
No vote yet
1 vote
Easy Math Editor
*italics*or_italics_**bold**or__bold__paragraph 1
paragraph 2
[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
Sort by:
Top Newest@KARAN SHEKHAWAT Here is the solution :
Let \(O\) represent the center of the polygon.
Now, \[\begin{align}\vec{XA_i}&=\vec{XO}+\vec{OA_i}\\ |\vec{XA_i}|^2&=|\vec{XO}|^2+|\vec{OA_i}|^2+2(\vec{XO} \cdot \vec{OA_i})\\ \sum_{i=1}^{n} |\vec{XA_i}|^2&=n(r^2+R^2)+2\sum_{i=1}^{n}(\vec{XO} \cdot \vec{OA_i})\\ &=n(r^2+R^2)+2\vec{XO} \cdot \left(\sum_{i=1}^{n} \vec{OA_i}\right)\end{align}\] 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.
Log in to reply
Awesome ! Thanks a lot ! It's is much better than complex number tecnique atleast for me :) You are genius !
Log in to reply
Can we use particular case of polygon in which vertex are n'th roots of of unity To prove it ?
Log in to reply
You could do that. We are to prove the general result which can also be, to my knowledge, proven that way.
Log in to reply
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 !
Log in to reply
Anyways I will upload it within a day or two.
Log in to reply
Well, I didn't solve it using complex numbers, I used vectors.
Log in to reply