Many questions have been posted on this site about the irrationality of $\pi$; I'll be referring to one such question here.
The accepted answer mentions that $\pi$ is irrational because it is the quotient of two numbers, namely the circumference and the diameter, with at least one of them being irrational. This makes complete sense to me, since the quotient of an irrational number with a rational one cannot be rational.
A rational number is a number that can be expressed as $p/q$, where $p$ and $q$ are integers. The number $\pi$ cannot be expressed in this form; hence it is irrational.
In other words, the definition of "fraction" does not include ratios like "circumference/diameter" in which the numerator and denominator are arbitrary numbers, not necessarily integers.
Now, later on in the answer, the author proves that the circumference or the diameter in a circle has to be irrational.
In the case of "circumference/diameter" (which you denoted $\pi = C/D$), it will always be the case that if the diameter is an integer, the circumference ($C = \pi D$) is not an integer, and if the circumference is an integer, the diameter ($D = C/\pi$) is not an integer: precisely because $\pi$ is irrational.
In this case, $\pi$ is being used to prove that $c$ or $d$ is irrational, and the irrationality of $c$ or $d$ is being used to prove that $\pi$ is irrational.
This above method does not makes sense to me; IMHO, you cannot prove A using B when you've proved B using A. In this case, I've read alternate proofs proving that $\pi$ is irrational, so I know that the circumference or diameter of a circle is also irrational.
However, are proofs like the above valid? Could I use B to prove A when I've proved B using A? Am I going wrong somewhere in my line of thinking?