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 Circle and Chord Properties
04-22-2010, 09:04 AM
Post: #1
 elim Moderator Posts: 577 Joined: Feb 2010 Reputation: 0
Circle and Chord Properties
Let $$OA \perp MN$$, $$P$$ and $$Q$$ are determined by $$B,C,D,E$$ on $$\odot O$$. Show that $$\overline{PA} = \overline{QA}$$
04-24-2010, 01:42 AM
Post: #2
 elim Moderator Posts: 577 Joined: Feb 2010 Reputation: 0
RE: Circle and Chord Properties

Let $$F$$ on $$\odot O$$ such that $$FC \parallel MN$$
Since $$B,C,F,E$$ are concyclic, $$\angle BEF +\angle BCF = \pi$$ and so one can easily see that
$$\angle PAF = \angle BEF$$ (left figure) or $$\angle PAF + \angle BEF = \pi$$ (right figure)
thus $$A,P,E,F$$ are concyclic and hence $$\angle PFA = \angle PEF = \angle QCA$$
also $$\angle PAF = \angle BEF = \pi - \angle BCF = \angle QAC$$, now $$AC = AF$$ implies that $$\triangle PAF = \triangle QAC$$

This is called "The butterfly theorem"
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