I think that my proof works out: Consider a red line, it may instersect a blue line on a blue point or a purple point. By the construction of that red line, it intersects at least four blue lines on a blue point, so there are at most $n-4$ purple points for each red line. It's easy to see that any four blue lines determines exactly three red lines. So we have $\frac{n(n-1)(n-2)(n-3)}{8}$ red lines, and then the answer is $\frac{n(n-1)(n-2)(n-3)(n-4)}{8}$ purple points. PS: Please tell me if i made any mistake.