TITLE: Semigroup law on points on the curve $f(x) = \frac{1}{x}$
QUESTION [6 upvotes]: Consider the positive half of the curve $f: \Bbb{R} \to \Bbb{R}, f(x) = \frac{1}{x}$.  Let $A = (a,1/a), B = (b, 1/b)$ be any two points on the curve.  Draw a line through them  Find where this point intersects the curve $f$ reflected across the $y$-axis.  Now reflect the point back across the $y$-axis to yield the result that we define to be $A + B$.  This is diagrammed below.

It appears to be an associative operation:

However, proving it to be associative seems way more involved than just drawing lines.  What's an easy route to prove that curve point addition is associative?
My attempt: There is positive $y$ such that $A + (B-A)t = (-y, 1/y)$ iff $a + (b-a)t = -y$ and $1/a + (1/b - 1/a)t = 1/y$ which leads to the quadratic equation: $\left( \frac{a-b}{a} - a(\frac{1}{b} - \frac{1}{a})\right )t + (\frac{1}{b} - \frac{1}{a})(a-b)t^2 -2 = 0$. This seems too complicated to work with on paper.

REPLY [2 votes]: modulo my unreliable arithmetic, the semi-group operation is:
$$
\theta_a \circ \theta_b = (\theta_a+\theta_b)\frac{\left(1+\sqrt{1+\frac{4\theta_a\theta_b}{(\theta_a+\theta_b)^2}} \right)}2
$$
which does not look too promising on the associativity front