Prove that you can choose three of spotlights.

Question

The plane is illuminated by several spotlights, each of which illuminates a half-plane. Prove that you can choose three of these spotlights, which also illuminate the entire plane?

Answer 1

One of the half planes must cover the final uncovered space (which is convex).

The convex shape it covers must have a closest point to the covering line (or a parallel in which case we can solve with two spotlights).

The covering line and the two lines radiating from the closest point cover the plane.

Answer 2

Rephrasing the other answer to make it more understandable (to me). The proof is by induction on $n$, the base case $n=3$ being obvious.

Remove one of the spotlights, which covers a half-plane $H$. If the remaining $n-1$ spotlights cover the plane, then you can select $3$ of them which cover the plane by the inductive hypothesis.

If not, then there will be an uncovered region which is a convex polygon. Let $P$ be a vertex of this polygon whose distance to $H$ is minimal, and let $e_1$ and $e_2$ be the edges adjacent to $P$. There is a spotlight whose boundary contains $e_1$, and points away from the uncovered region, therefore pointing towards the complement of $H$. Same goes for $e_2$. Therefore, some thought should convince you that $H$, together with the spotlights for $e_1$ and $e_2$, will cover the whole space.