Sphere-plane intersection - Shortest line between sphere center and plane must be perpendicular to plane?

Question

Problem

Let's say we have a sphere $S$, and an intersecting plane $P$, creating a circle (i.e. the intersection is not just tangential), the following quote arises in my textbook:

The shortest distance between the center $C$ of sphere $S$ and the plane $P$ must always be perpendicular to the plane itself.

Proof?

Intuitively this makes perfect sense, and drawing it in 2D with a cirle-line intersection validates the point.

But my textbook offers no proof of this statement, and it's a bit frustrating.

I can calculate the shortest point-plane distance using a foruma, but I haven't been able to use this to confirm the above-mentioned statement.

Reduction of problem

I realize that the sphere part is obsolete, as the argument can be reduced to the idea that any shortest point-to-plane vector must be perpendicular to the plane. (Right?)

Question

Are there any nifty proofs for this "shortest point-plane vector must be perpendicular to plane" statement?

Answer 1

Let $H$ be the intersection between the plane and the perpendicular to it passing through point $P$. If $Q$ is any other point on the plane, let's show that $PH<PQ$. In fact, triangle $PHQ$ has a right angle at $H$ (by definition of perpendicular), hence its hypotenuse $PQ$ is greater than leg $PH$, Q.E.D.