Proof for tangent chord angle formula

Question

I want a proof for tangent chord angle formula by using the following method: Drawing a parallel line - See the diagram. I know the other proofs and I want to prove it with drawing a parallel line.

Answer 1

If $AB \parallel CD$, then $\angle BAC=\angle ACD$.

Because $AB$ is tangent (and $AB \parallel CD$), then $|AC|=|AD|$, so the triangle $ACD$ is isosceles and $\angle ADC=\angle ACD$.

$\angle AOC$ is central angle with inscribed angle $\angle ADC$, so $$(\widehat{AC}=)\angle AOC = 2 \angle ADC = 2 \angle ACD = 2 \angle BAC$$ We have then $$\frac{\widehat{AC}}{2}=\angle BAC$$