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A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
The distance from the center of a circle, which is tangent to a ellipse $x^2/a^2+y^2/b^2=1$ and two parallel tangent lines of the ellipse is $a+b$
Consider the following problem: Let $E$ be the ellipse $x^2/a^2+y^2/b^2=1$ with $a>b$. Consider two tangent lines on $E$ which are parallel, say, $r$ and $s$. Let $C$ be a circle, which is ...