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The maximum number of points in a plane such that the distance of any of these points from a given point in the plane is less than the distance of it from any other point is five.

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What is your question? Do you want a proof, a reference, the name, a generalisation, an application? Please edit your post to make it clear exactly what it is you are looking for. – Michael Albanese Jan 5 '13 at 3:32

Let $O$ be your given point.

Let the points $P_i$ satisfy your conditions, i.e. that $OP_i < P_i P_j \, \forall i, j$

Use Cosine rule to show that $\angle P_i O P_j > 60^\circ\, \forall i, j$.

Now order points $P_i$ in a clockwise manner around $O$. It follows that there are $< \frac {360}{60}$ such points. Hence, there are at most 5.

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