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.
Proof for tangent chord angle formula
2
$\begingroup$
geometry
euclidean-geometry
circles
angle
-
0I'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 – 2017-01-12
1 Answers
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 $$\frac{\widehat{AC}}{2}=\angle BAC$$
-
0Hi. 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. – 2017-01-13
-
0@titansarus - see [inscribed angle](https://en.wikipedia.org/wiki/Inscribed_angle) and [Alternate interior angles](https://www.mathsisfun.com/definitions/alternate-interior-angles.html) – 2017-01-13