20.27. On a plane, there is a finite set of polygons (not necessarily convex), each two of which have a common point. Prove that there exists a line that has common points with all these polygons.
Solution
20.27. Let's take an arbitrary line in the plane and project all polygons onto it. In this case, we will obtain several segments, any two of which have a common point. Consider the left ends of these segments and choose the rightmost one (to make it clear what "right" and "left" mean, a direction needs to be defined on the line). The resulting point belongs to all segments, so a perpendicular line drawn through it to the line intersects all the given polygons.
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