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Angle Ratio Ascertainment 3 (Posted on 2016-05-22) Difficulty: 3 of 5
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    
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Solution Solution Comment 3 of 3 |

Let A be the intersection of line PQRS
and the common chord EF (excluding the
points 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.

QED

Note: /XYZ denotes angle XYZ and
      ^XYZ denotes triangle XYZ.
      

Edited on May 22, 2016, 8:21 pm
  Posted by Bractals on 2016-05-22 20:19:56

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