Are triangles the strongest shape? They are according to the buzz on the Internet (and most stable too), despite competition from circles. Mythbusters even proclaim that "triangles are the strongest shape because any added force is evenly spread through all three sides". 
Is there a way to make some precise sense of the question, and if so, how does one actually prove that triangles are the "strongest"?
 A: Here's one part of it.  
As far as polygons go, a triangle is the only one that is defined by its side lengths.  If you have a triangle of sides 5,6, and 7, there is only one shape it can take.  The same cannot be said of other polygons.  Imagine a square.  It can be squished into a diamond with the same side lengths.  
There is SSS congruence for triangles, but no analogous congruence for other polygons. 
That's what diagonal bracing does in physical structures.  Creates triangles.
A: As you asked about the strength of a triangular shape then let me introduce to the triangular chain consisting of three rigid links or bars connected to each other by pin joints(allowing rotation between two joined links) .  
The degree of freedom (n) of a plane chain is given by the Grasshoff's law as $$n=3(l-1)-2j-h$$ for a triangular chain we have $$l=\text{no. of links}=3$$
$$j=\text{no. of binary joints}=3$$
$$h=\text{no. of higher pairs}=0$$ Hence, we get $$n=3(3-1)-2(3)-0=6-6=0$$ The degree of freedom of the triangular chain (equivalent to plane triangular shape) has zero degree of freedom this indicates that links of the triangular chain can't move even a bit if links are strong enough even under the application of external forces. 
Thus a triangular shape is the strongest one which is also called a rigid structure. It is also called a perfect frame in physical structures. 
A: If the links are rigid (i.e. can't be deformed) then there will be no relative motion between any two pin-jointed links of a triangle while quadrilateral & other polygons have degree of freedom $\ge 1$ hence other polygons can be made to out of shape by applying force so triangles  are only rigid or the strongest shapes.   
