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Seeking Sums With Inradius And Exradius (Posted on 2007-10-08) Difficulty: 3 of 5
Triangle PQR is equilateral with PQ = QR = RP = 2. The line QP is extended to meet at point S such that P lies between S and Q.

Y is length of the inradius of Triangle SPR while Z is the length of the exradius of Triangle SQR with respect to the side QR.

Determine Y+Z.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 2 of 4 |

Let I be the incenter of triangle SPR and A
the point of tangency of the incircle with
side SP. Then
          |SP|+|PR|+|RS|
  |PA| = ---------------- - |RS|
                2
          |SP|-|RS|+|PQ|
       = ----------------
                2
Let E be the excenter of triangle SQR (with
respect to side QR) abd B the point of
tangency of the excircle with side SQ extended.
Then
          |SQ|+|QR|+|RS|
  |QB| = ---------------- - |SQ|
                2
          |RS|-|SP|
       = -----------
              2
Therefore,

  Y+Z = |AI|+|BE| = (|PA|+|QB|)*sqrt(3)
         |PQ|
      = ------ * sqrt(3)
           2
Which is the altitude of triangle PQR.
For our problem,
  Y+Z = sqrt(3)

    

  Posted by Bractals on 2007-10-08 14:26:24
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