Power of a point

Here’s a simple proof: Let $\vec{\alpha }$ be a unit vector denoting the direction of our chord, and $\vec{P}$ be our position. Then, the desired lengths are the signed values of $t$ such that $\Vert P+t\alpha \Vert =r$. Expanding, we get that

$$P^2+2t(P\dot{\alpha })+t^2=r^2$$

By Vieta’s, the products of the roots of this equation are just $P^2-r^2$, which, importantly, is independent of $\alpha$. This quite elegantly proves Power of a Point without needing to resort to casework.