The incircle of the triangle PQR touches its sides QR, RP and PQ respectively at the points S, T and U. It is known that the respective lengths of QS, RT and PU (in that order) are consecutive integers and the length of the inradius of of the triangle PQR is 4 units.

Determine the respective lengths of each of the sides PQ, QR and RP of the triangle.

Let the side lengths be p, q, and r and the
semiperimeter s = (p+q+r)/2.

  |QS| = k-1 = s-q
  |RT| =   k = s-r
  |PU| = k+1 = s-p

Adding these gives

  3k = 3s-(p+q+r) = 3s-2s = s

  Area(PQR) = inradius*semiperimeter

            = sqrt(s(s-p)(s-q)(s-r))

            or

  16s^2 = s(s-p)(s-q)(s-r)

            or

  16(3k) = k(k^2 - 1)

            or

  k = 7

Therefore,

  |QS| = 6
  |RT| = 7
  |PU| = 8

and

  |PQ| = |PU|+|UQ| = |PU|+|QS| = 8+6 = 14 
    
  |QR| = |QS|+|SR| = |QS|+|RT| = 6+7 = 13

  |RP| = |RT|+|TP| = |RT|+|PU| = 7+8 = 15

Posted by Bractals on 2007-12-06 11:43:01