**Original problem**

If 8 points in a plane are chosen to lie on or inside a circle of diameter 2cm then show that the distance between some two points will be **less than** 1cm.

**My proof**

Let the points be $p\_1,p\_2,\dots,p\_8$

Placing point $p\_8$ at the center of the circle and placing other seven points on the periphery of the circle, at equal distances.(remember we are required to prove that there are at least a pair of points whose distance from each-other is less than 1cm.)

Now we have one point at the center and seven at the periphery of the circle.

The heptagon which will be formed by joining the seven points on the circle will have all its side equal in length.

The length of the side of the heptagon is less than $\frac{2\pi}{7}$cm which is less than 1cm.

Now if we replace the position of any point then too we get a pair of points whose distance from each-other is less than 1cm.

**End of proof**

I think that I have proved it correctly. But my brother was saying that this is just a verification because you are choosing the points by yourself. 
I think that this is a **in general** proof rather a particular verification.

**Is my proof correct in general or not? If not then please provide an in general proof?** 

**EDIT**(Another proof)

I was thinking of another technique. In proof by contradiction, for example proving $\sqrt{2}$ is irrational, we first assume that $\sqrt{2}$ is rational and then we use the property of rationals and find that some properties is not true for $\sqrt{2}$, then this contradicts and we say that $\sqrt{2}$ is irrational.

**In this proof** firstly I'm making a unit circle(with center O). Taking a point on the periphery we draw a circle of radius 1cm, now we say that no other points will lie in the shaded region.

Distance between $p\_1$ and $p\_2$ is exactly 1cm.
![enter image description here](https://i.stack.imgur.com/uF1uE.jpg)

I'm placing a point($p\_1$) on the periphery and taking that no other point lie at a distance of less than 1cm from $p\_1$. I'm repeating this thing for $p\_2$ and $p\_3$. If we do this for the fourth point($p\_4$), then after that we can see that the unshaded region is very small, from here we can easily show that the separation between a pair of points is less than 1cm.

**The reason why I have taken all points on the periphery and not inside the circle.**
The reason is that, because when we take any point inside the circle then the *area covered by that point*\* inside the circle will be more as compared to when we take that point on the periphery.

\*area covered by that point - I mean the area covered by the unit circle with center as that point. 

**Is this proof correct or not? Is this equivalent to the proof by contradiction or not?**