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Dirichlet theorem in geometry

Dirichlet are mostly used in geometric problems where problem solvers have to divide their figure into different parts in order to make their own cages. Let’s get a deeper look into some problems: 

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Another problem can be mentioned below (not a famous one, but is a very useful example): 

In a 1-side square there are some circles. The total perimeter of all these circles is 10. Prove that there exists a line that goes through at least 4 circles.

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