Intersections of lines from adding points on an elliptic curve

I was playing with a graphing program this morning and tried graphing an elliptic curve and all of the lines that you'd use to add the points on the curve with integer coordinates using the cord-and-tanget method of addition. Here's what I got for the elliptic curve y² = x³ + 1.

There must be something interesting that you can state and prove about the intersections of those lines.