APMO 2015 - Asian Pacific Mathematics Olympiad Question 4

Let $n$ be a positive integer. There are $2n$ distinct lines in the plane, no two parallel, of which $n$ are blue and $n$ red. Let $B$ be the set of points on at least one blue line and $R$ the set of points on at least one red line. Prove that a circle exists meeting $B$ in exactly $2n-1$ points and $R$ also in exactly $2n-1$ points.

Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand.

Example arrangement (new):