Draw a Circle

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Naeem588
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Draw a Circle

Unread post by Naeem588 » Sat Jun 05, 2021 11:32 pm

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.

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Anindya Biswas
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Re: Draw a Circle

Unread post by Anindya Biswas » Mon Jun 07, 2021 1:39 pm

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

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emeryhen121
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Re: Draw a Circle

Unread post by emeryhen121 » Wed Sep 15, 2021 12:45 pm

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.

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