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.
Best Answer
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 $47$.