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.

I understand that a point $P$ is chosen at random inside a triangle $ABC$ according to a uniform probability distribution, and you want the expected value of the sum of the distances from $P$ to the sides of the triangle.

The distance from $P = (x,y)$ to one of the sides is a linear function $ax + by + c$ of the coordinates $x, y$. Thus the sum of the distances is also linear. Therefore the average value is the average of the values for $P = A$, $P = B$ and $P = C$, i.e. the average of the three altitudes of the triangle.

In the present case the altitudes are $36, 45, 60$. So the expected value is $47$.