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Equal Angles (Posted on 2005-12-14) Difficulty: 3 of 5
Let circle A be in the interior of circle B and tangent to it at point M. Let chord QR of circle B be tangent to circle A at point P. Prove that angles PMQ and PMR are equal.

1993 British Mathematical Olympiad,Round 1,Problem 4.

  Submitted by Bractals    
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Solution: (Hide)
Let circles A and B have radii a and b respectively.
Letting M be the origin and using polar coordinates
the circles have the following equations:

   Circle A:  r = 2a cos(theta)

   Circle B:  r = 2b cos(theta)

where theta is the angle between r and the line
determined by the centers of the circles.

Let Q' be the intersection of MQ and circle A.
Let R' be the intersection of MR and circle A.

   MQ'     2a cos(theta_q)     a     2a cos(theta_r)     MR'
  ----- = ----------------- = --- = ----------------- = -----
   MQ      2b cos(theta_q)     b     2b cos(theta_r)     MR

  ==> triangles Q'MR' and QMR are similar

  ==> Q'R' and QR are parallel

  ==> AP and Q'R' are perpendicular

  ==> <PAQ' = <PAR'

  ==> <PMQ = <PMQ' = (<PAQ')/2 = (<PAR')/2 = <PMR' = <PMR

Comments: ( You must be logged in to post comments.)
  Subject Author Date
To elaborateDrBob2005-12-15 10:44:19
re: re is there an easier wayDrBob2005-12-15 10:42:25
Questionre: Is there an easier way? ???Jer2005-12-14 14:28:39
Is there an easier way?DrBob2005-12-14 12:15:09
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