Ten points are chosen on a circle. A line segment is drawn between every pair of chosen points.
Every time two line segments intersect inside the circle, that intersection point is marked red. What is the largest number of points that can be marked red?
For example, for five points on the circle, we can get up to five red intersection points.
![[asy] unitsize(2 cm);pair[] A;pair P;int i, j;A[1] = dir(18);A[2] = dir(60);A[3] = dir(124);A[4] = dir(190);A[5] = dir(330);A[6] = A[1];A[7] = A[2];A[8] = A[3];A[9] = A[4];draw(Circle((0,0),1));for (i = 1; i <= 5; ++i) { dot(A[i]);for (j = i + 1; j <= 5; ++j) { draw(A[i]--A[j]);} } for (i = 1; i <= 5; ++i) { P = extension(A[i], A[i + 2], A[i + 1], A[i + 3]);dot(P,red);} [/asy]](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2F4%2F1%2F7%2F41744a953a4828e81976b3a11c5522b232e7aa6b.png)
