## Draw a Circle

There are 2019 points in a plane. Prove that a circle be drawn in such way that 1009 points lie Outside and inside the cricle respectively. And the last point is on the circumference.

- Anindya Biswas
**Posts:**263**Joined:**Fri Oct 02, 2020 8:51 pm**Location:**Magura, Bangladesh-
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### Re: Draw a Circle

There are only finitely many lines that goes through at least $2$ of the points. Let's choose a line $l$ not parallel to any of them. So, $l$ goes through at most one point in that plane. Now translate the line $l$ such that it goes through a point in that plane and the number of points on both sides of the line is the same. Let's call this point $P$.

Now draw a circle tangent to $l$ at point $P$. Increase it's radius until it encloses all $1009$ points of one side of the line $l$.

Now draw a circle tangent to $l$ at point $P$. Increase it's radius until it encloses all $1009$ points of one side of the line $l$.

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

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**John von Neumann**- emeryhen121
**Posts:**12**Joined:**Fri Jul 16, 2021 6:04 pm

### Re: Draw a Circle

This is how a circle be drawn that 1009 points lie Outside and inside the cricle respectively with the last point being on the circumference:

Choose a line l, there are only finitely many lines that goes through at least 2 of the points but this line is not parallel to any of them. So, l goes through at most one point in that plane. Presently interpret the line l to such an extent that it goes through a point in that plane and the quantity of focuses on the two sides of the line is similar. Let's call this point P. Next, draw a circle tangent to l at point P. Increase it's radius until it encloses all 1009 points of one side of the line l.

Choose a line l, there are only finitely many lines that goes through at least 2 of the points but this line is not parallel to any of them. So, l goes through at most one point in that plane. Presently interpret the line l to such an extent that it goes through a point in that plane and the quantity of focuses on the two sides of the line is similar. Let's call this point P. Next, draw a circle tangent to l at point P. Increase it's radius until it encloses all 1009 points of one side of the line l.