Are triangles the strongest shape?

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.

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.