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Find the Ratio (Posted on 2007-08-20) Difficulty: 2 of 5
I (radius=r) is the incircle of triangle ABC (right angled at B) and O (radius=R) is another circle which touches the extended side BC and AC. Find the ratio (R/r) if the point of intersection of circles I and O is midpoint of AC.

See The Solution Submitted by Praneeth    
Rating: 3.2500 (4 votes)

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Solution Solution | Comment 1 of 3

If the incircle of triangle ABC touches the
midpoint of the hypotenuse, then ABC is a
isoceles right triangle.
Let L and M be the points where the incircle
touches sides AB and AC rspectively.
       |AC|^2 = |AB|^2 + |BC|^2
    (2|AM|)^2 = 2|AB|^2
      4|AM|^2 = 2(|AL| + |LB|)^2
              = 2(|AM| + r)^2
             or
            0 = 2|AM|^2 - 4r|AM| - 2r^2
             or
         |AM| = r(1 + sqrt(2))
From similar triangles OMA and AMC we get
        |OM|     |AM|
       ------ = ------
        |AM|     |IM|
             or
           rR = |AM|^2
             or
           R
          --- = 3 + 2sqrt(2) ~= 5.828427
           r        
 

  Posted by Bractals on 2007-08-20 11:23:46
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