What is the average sum of distances of a random point inside a triangle to its three sides? Given a non- Equilateral Triangle with following side sizes: $45,60,75$.
Find the sum of distances from a random located point inside a triangle to its three sides.
Note 1: Viviani's theorem related only to equilateral triangles.
Note 2: Fermat point is related to the minimization of distances from a random point inside the triangle and its vertices.
As we can see that both notes are not helpful to solve that problem.
I have been given that puzzle during an hour an a half exam. There were only 6 minutes to solve that problem.
Afters many hours I still do not have an answer.
I will be very glad to get some assistance or maybe the whole solution
Regards,
Dany B.

Let $ (x,y)$ be a point in a semi-circle with diameter inclined at $ \sin ^{-1} \frac35 $ to x-axis having sides proportional to Pythagorean  triplet (8,6,10) as given. 
The three perpendicular distance to sides of a scalene triangle are $ (x,y, 4 x/5 + 3 y/5) , $
totaling to  $ \dfrac {9 x + 8 y} {5}   $ which is variable , needing to be averaged.
If it were constant, a result like  those from Viviani, Fermat would have been in existence now for more than two centuries.
