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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.enter image description here

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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 $$\frac{\widehat{AC}}{2}=\angle BAC$$

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  • $\begingroup$ Hi. How did you get the first part? I know one proof for the first part but it uses the fact that $\angle BAC = \widehat {AC} /2$ which is what we want to prove in the question. $\endgroup$ – titansarus Jan 13 '17 at 5:29
  • $\begingroup$ @titansarus - see inscribed angle and Alternate interior angles $\endgroup$ – Jaroslaw Matlak Jan 13 '17 at 5:50

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