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Incircle Points (Posted on 2015-11-04) Difficulty: 3 of 5
Consider a triangle PQR where PQ = 2014, QR = 2015, RP =2016. RM is an altitude of this triangle.

Determine the distance between the points at which the incircles of PRM and QRM touch RM.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1

Let a = PM, r-a = MQ, h = RM, p = QR, and q = RP.
Applying the Pythagorean theorem to triangles
PMR and QMR gives
   h^2 = q^2 - a^2 = p^2 - (r-a)^2
              ==>
   2a = (q^2 + r^2 - p^2)/r
The distance between tangency points is
       |  a + h - q     (r-a) + h - p  |
   d = | ----------- - --------------- |
       |      2                2       |
       |  p - q - r + 2a  |
     = | ---------------- |
       |         2        |

       |  p - q - r + (q^2 + r^2 - p^2)/r  |
     = | --------------------------------- |
       |                  2                |
     = | q(q - r) - p(p - r) |/(2r)
For our problem,
   d = | 2016(2016 - 2014) - 2015(2015 - 2014) | / 4028
     = 2017/4028 ~= 0.5007447864945382323733862959285
QED

Edited on November 4, 2015, 2:07 pm
  Posted by Bractals on 2015-11-04 14:05:04

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