Suppose a homogeneous triangle with mass $M$ is in equilibrium hanging by 3 strings from its vertices the ceiling 
Suppose a homogeneous triangle with mass $M$ is in equilibrium hanging by 3 strings from its vertices the ceiling; show that the tension of each string is proportional to its length.

Here is what I tried, and I'm almost sure that this leads to a solution. However I don't know how to proceed. Also, I'd like to know if there was a more elegant solution. Any help is appreciated, but I prefer tips and insights rather than full solutions.
 A: Since you asked for hints, I will just give some insights:
a) You need not deal with breaking them to $X,Y,Z$ components. If a proof of a general statement is asked, most of the time it can be done with abstract vectors.
b) If you chose the common point of the 3 strings on the ceiling as origin, calculations will be a lot more easier to do, as you get both the position vectors of the 3 vertices and the directions of the 3 tension forces at the same time. (since it can only act along the string)

 c) For this question you will need to prove that the position vector of the centroid is also along direction of gravity (i.e. vertically downward.)


 d)You will need to use scalar triple product for this one, if you don't know this beforehand, here is the introduction for it which is good enough for this question.


 e) Like lami's theorem there is a similar concept for 4 vectors which is an key idea to use in equilibrium related questions, here is the concept.

if you still need help in solving the question, here is the solution
