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.
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titansarus
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I'm not sure what other proofs you refer to but is it not the same as http://math.stackexchange.com/questions/12885/angle-between-chord-and-tangent?rq=1 – John Wayland Bales Jan 12 '17 at 22:01
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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 $$\frac{\widehat{AC}}{2}=\angle BAC$$
Jaroslaw Matlak
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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. – titansarus Jan 13 '17 at 05:29
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@titansarus - see inscribed angle and Alternate interior angles – Jaroslaw Matlak Jan 13 '17 at 05:50