Multiplication using Parabola

Theorem
Let $\map f x = x^2$.

Let $A = \tuple {x_a, y_a}$ and $B = \tuple {x_b, y_b}$ be points on the curve $\map f x$ so that $x_a \le x_b$.

Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$.

Proof
Let $\map f x = x^2$.

Then:


 * $\map f {x_a} = x_a^2$

and:


 * $\map f {x_b} = x_b^2$

Then the slope of the line segment joining $A B$ will be:

The result follows from taking the equation of the line defined by its slope and either point $A$ or $B$ to calculate the $y$-intercept.

Proceeding with point $A$: