Problem 4
Let be a positive integer. Consider distinct lines on the plane, no two of which are parallel. Of the lines, are colored blue, the other are colored red. Let be the set of all points on the plane that lie on at least one blue line, and the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects in exactly points, and also intersects in exactly points.
Step 3 of 4: Construct the circle
Detailed analysis
Take the circle C tangent to the right side of R and tangent to both selected lines. Each other line meets both selected lines inside R. It cannot cross the two horizontal sides of R, since that would give a forbidden direction between the selected directions; hence it crosses the two vertical sides.