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Circle and Chord Properties
04-22-2010, 09:04 AM
Post: #1
Circle and Chord Properties
Let \(OA \perp MN\), \(P\) and \(Q\) are determined by \(B,C,D,E\) on \(\odot O\). Show that \(\overline{PA} = \overline{QA}\)    
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04-24-2010, 01:42 AM
Post: #2
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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