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Perpendicular Incenters (Posted on 2018-05-06) Difficulty: 3 of 5

  
Let a, b, and c be the lengths of sides BC, CA, and AB respectively
of ΔABC. Let D be a point on side BC. Let I and J be the incenters
of ΔABD and ΔACD respectively.

If the line segment IJ ⊥ AD, then what is the value of the ratio
|BD| / |CD| in terms of a, b, and c?
  

  Submitted by Bractals    
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Solution: (Hide)

  
Let E and F be the points on AD where the incircles of ΔABD and
ΔACD kiss AD. Clearly IJ ⊥ AD when E and F coincide.

                              |DE|   =   |DF|

         |BD| + |AD| - |AB|        |CD| + |AD| - |AC|
        ----------------------   =   ----------------------
                     2                                     2

        |BD| = ( |AB| + |BC| - |AC| ) / 2 = (c + a - b) / 2.

   Note: D is the point on BC where the incircle of ΔABC kisses BC.

        |CD| = (a + b - c) / 2

        |BD| / |CD| = ( c + a - b ) / ( a + b - c )

QED
  

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