[betterunix2]

"If you put 2 and 2 together, you get 4, a toddler can see that, but how on earth did anyone arrive at the conclusions that (-2.0, 1.4) + (1.9, 2.3) = (0.1, -1.9) via drawing a line, finding a third point, then reflecting across the X-axis? Makes no sense to me at all."

Actually, that is the simplified version of the group law which is not what is actually "derived" in the theory. To derive the group law you work with rational functions defined on the elliptic curve (i.e. defined on the coordinates of points on the curve). The group itself is actually the "divisor class group" of the curve, which you can read about (but it is fairly advanced material).