Geometrical constructions? If you have an accurate visual intuition of the properties of triangles and squares, then a diagram of the Pythagorean theorem pretty much is a proof of it. Euclidean geometry is an axiomatic system, but it isn't a formal system on strings of symbols; it's a formal system on abstract geometric figures.
Throughout most of history, math wasn't done in what we now think of as "mathematical notation", i.e. expressions written symbolically. That wasn't invented until the 1500s. Before then, math was done with proofs written in ordinary language, accompanied by diagrams.
Before language, people must have thought without words. I often have the impression that I have a thought fully-formed in my head, yet I wait to listen to it unfold in words before moving on to the next thought. Perhaps I could think much faster if I weren't addicted to words.
Has anyone developed techniques for thinking without words?
This would have a little in common with Buddhist practices of emptying your mind, but wouldn't be the same thing. For one thing, Buddhists also try to empty their minds of images. More importantly, they are trying not to think, while I'm trying to think - just not unpack everything into words.