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## Halfplanes cover

Suppose that the plane can be covered by $n$ half-planes. Then we can find three of these half-planes which cover the plane.

Proof: Suppose $A_i$ are the given half-planes and $B_i$ are their complements. The condition $\displaystyle \bigcup A_i = \mathcal{P}$ is equivalent to $\displaystyle \bigcap B_i=\emptyset$. If no three half-planes cover the plane then every three of the $B_i$ intersect. Since $B_i$ are convex sets, it follows from Helly’s theorem that their intersection is not void. Contradiction.
Therefore, we can find three of the given half-planes which cover $\mathcal{P}$.