Let, $ P_1P_2P_3\ldots P_n$ be a regular polygon whose circumcentre is $O$. Prove that,

\[\sum_{i=1}^n \overrightarrow{OP_i}=0\]

## Vectors around Regular Polygon

### Re: Vectors around Regular Polygon

We choose the coordinate system so that $\overrightarrow{OP_1}$ is parallel to $x$-axis and pointed to the positive direction. Breaking each vector into components we get $\overrightarrow{OP_i}=r\cos\dfrac{2\pi(i-1)}{n}\hat{\mathbf{i}}+r\sin\dfrac{2\pi(i-1)}{n}\hat{\mathbf{j}}$ so \[\sum_{i=1}^n \overrightarrow{OP_i}=r\sum_{i=0}^{n-1}\cos\dfrac{2\pi i}{n}\hat{\mathbf{i}}+r\sum_{i=0}^{n-1}\sin\dfrac{2\pi i}{n}\hat{\mathbf{j}}.\] The proof that the first sum (and the second analogously) is equal to $0$ is proven in example $2$ here.

**- What is the value of the contour integral around Western Europe?**

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.

### Re: Vectors around Regular Polygon

\[\begin{align*}

2\sum_{i=1}^n \overrightarrow{OP_i} &= \sum_{i=1}^n (\overrightarrow{OP_i} + \overrightarrow{OP_{i+2}} )\\

&= k \sum_{i=1}^n \overrightarrow{OP_{i+1}}

\end{align*}\]

Where $k < 2$ as $\overrightarrow{OP_i} $ and $\overrightarrow{OP_{i+2}} $s are not in the same line.

So, \[(2-k)\sum_{i=1}^n \overrightarrow{OP_i} = 0 =\sum_{i=1}^n \overrightarrow{OP_i}\]

2\sum_{i=1}^n \overrightarrow{OP_i} &= \sum_{i=1}^n (\overrightarrow{OP_i} + \overrightarrow{OP_{i+2}} )\\

&= k \sum_{i=1}^n \overrightarrow{OP_{i+1}}

\end{align*}\]

Where $k < 2$ as $\overrightarrow{OP_i} $ and $\overrightarrow{OP_{i+2}} $s are not in the same line.

So, \[(2-k)\sum_{i=1}^n \overrightarrow{OP_i} = 0 =\sum_{i=1}^n \overrightarrow{OP_i}\]

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Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

### Re: Vectors around Regular Polygon

Two words:

"Everything should be made as simple as possible, but not simpler." - Albert Einstein

### Re: Vectors around Regular Polygon

Hmm, Nayel Vai... Complex Numbers definitely give a straightforward solution. But, Mahi's approach was very neat.

Solution using complex numbers:

Solution using complex numbers: