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 Angle Ratio Ascertainment 3 (Posted on 2016-05-22)
Two circles meet at E and F.

A line meets the circles at points P,Q, R and S in that order.

Find : ∠PEQ/∠RFS

 No Solution Yet Submitted by K Sengupta No Rating

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 Solution Comment 3 of 3 |

`Let A be the intersection of line PQRSand the common chord EF (excluding thepoints E and F).Equal Inscribed Angles:   /APE = /RPE = /RFE = /RFA         (1)   /ASF = /QSF = /QEF = /QEA         (2)Vertical Angles:   /EAQ = /EAP = /FAS = /FAR         (3)Similar Triangles:         (1)&(3) ==> ^EAP ~ ^RAF           (4)   (2)&(3) ==> ^FAS ~ ^QAE           (5)Proof:   (4) ==> /PEA = /FRA   /PEA = /PEQ + /QEA                (6)   /FRA an external angle of ^FRS        ==> /FRA = /SFR + /RSF                 = /RFS + /QSF       (7)   Combining (2), (6), and (7) gives      /PEQ = /RFS   Therefore, the ratio is one.QEDNote: /XYZ denotes angle XYZ and      ^XYZ denotes triangle XYZ.`
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Edited on May 22, 2016, 8:21 pm
 Posted by Bractals on 2016-05-22 20:19:56

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