I studied mathematics and I was taught to draw pictures if problem is too hard to solve. What kind of tools/programming languages there are aimed to draw pictures for math problems? Should one learn Tikz or some programming language?

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The range of software in this category is quite vast. You will need to do some research to find items that satisfy your precise needs.

I have used https://www.desmos.com/ to make drawings in two-space. You can use https://alternativeto.net/software/desmos/ to investigate related possibilities. And this site may help you to find something like Tikz.

I think it's reasonable to say that, most open source mathematics systems have some form of graphics system for displaying various mathematical entities. Look for pages like https://en.wikipedia.org/wiki/List_of_open-source_software_for_mathematics for possibilities.

To answer your question directly, yes, you could learn Tikz. Alternatively, you could learn Python or Ruby. I am familiar with the former language and know that many products are available for it for manipulating mathematical objects. It's not difficult to learn the rudiments of it.

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