Using a Parabola as a Calculator

Alright so I was watching this cool video on YouTube about the awesome proofs that Archimedes did about parabolas.

And I was immediately nerd sniped by this amazing claim, a parabola can be used as a calculator.

So for any two numbers x and y the quantity x×y is the y-intercept of the line segment from (-y, y²) to (x,x²).

WOAH! That’s fascinating. Alright, let’s prove it.

Remember high school geometry? Well, if you recall your old y = mx+b you’ll recall the slope is rise over run, so the slope must be

This happens to be a difference of squares so we can simplify it to

Now if you recall the point slope form we can work out the equation of the line, since we’ve already assigned x and y let’s call this function g(i)

We can choose either (x,x²) or (-y,y²) to plug in, I decided to use x. Which when solving for g(i) indeed produces, as expected

So it is indeed true, you can take any two numbers on the x-y plane, draw a parabola f(x)=x² and do multiplication just by drawing a line connecting the two points on the parabola represented by x² and y².