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Ray Ratio (Posted on 2011-10-23) Difficulty: 2 of 5
Circles C1 and C2 intersect at points A and B. Points P1 and Q1 on circle C1 and points P2 and Q2 on circle C2 are such that P1P2 and Q1Q2 are common tangents to circles C1 and C2. Ray Q1B is parallel to P1P2 and intersects circle C2 again at point R2.

What is the value of |BR2|/|BQ1|? Prove it.

  Submitted by Bractals    
Rating: 5.0000 (1 votes)
Solution: (Hide)

Let O1 [O2] be the center of circle C1 [C2].
Let line O1P1 [O2P2] intersect ray Q1B at
point S1 [S2].

     O1P1 [O2P2] ⊥ P1P2 and P1P2 || ray Q1B

   ⇒ O1P1 [O2P2] ⊥ ray Q1B

   ⇒ S1 [S2] bisects chord Q1B [BR2]

Let x = |Q1S1| = |S1B| and y = |BS2| = |S2R2|.

The common tangents have the same length, thus 

    |Q1Q2| = |P1P2| = x+y.

The circle power of point Q1 with respect to
circle C2 gives

   |Q1Q2|2 = |Q1B|•|Q1R2|

   (x+y)2 = 2x⋅2(x+y)

   x+y = 4x

   y = 3x

Therefore,

    |BR2|     2y     6x
   ------- = ---- = ---- = 3
    |BQ1|     2x     2x

QED

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsPossible solutionbroll2011-10-24 01:10:47
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