Yesterday I read up on Diophantine equations and the property in which two rational solutions of such an equation make a line that gives a third rational solution to the equation. I was thinking about pairs of rationals where this property isn't especially obvious to someone new to the subject. I noticed that if you have the equation $y^2 = \frac{1}{3}x^3 - 2x$ and you have two rational solutions that make a vertical line (or a line with a very large slope) that you might get something like in this illustration where the red curve is the Diophantine equation and the dashed line is a line created by two rational solutions...
If the dashed line could represent the line created by two solutions the property in question wouldn't work so clearly it can't happen. But why? Similarly why couldn't a similar situation with a non vertical line with a large slope not happen?
This is a partial answer. According to https://en.wikipedia.org/wiki/Elliptic_curve a vertical line means for the points P and Q that we use to get that vertical line our P = -Q and our third homogenous coordinate that intersects the Diophantine equation is just the additive identity. But it seems like there could be some steeply slopped lines for which the line is not a tangent point to the curve for P or Q but who doesn't intersect the curve at a third homogenous coordinate R. What's going on in situations like this?