Mean Perimeter:   3-4-5 Triangle Case

    In 3-4-5 triangle with legs 3 and 4 and hypotenuse 5,
  inscribe the arbitrary triangle DEF.
  Assuming that vertices D, E, and F are randomly
- with uniform probability density -
  distributed, respectively, inside the BC=3, AC=4, and AB=5,
  find the mean perimeter of the triangle DEF.

        Answer:
    (20460 + 9728 Ln[2] + 5103 Ln[3])/4500

        Numerically:
     7.29092317368011371671128233812462