I am writing a math exam and I'd like to include a picture. I want a free drawing program so that I can draw the coordinate axes with a unit circle. Inscribed in the unit circle will be a hexagon. The interior of the hexagon will be divided into triangles. Which drawing program will make this process as painless as possible?
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You can see a good list at Wikipedia.
I suggest you try GeoGebra, which is convenient for geometric constructions (for more general things, Inkscape is very good).
Since you mention nowhere that your program has to include a GUI, TikZ is always mentionable (although I know that if this is just a quick assignment for an exam, even using LaTeX might seem like overkill).
You can look at the official documentation here: paws.wcu.edu/tsfoguel/tikzpgfmanual.pdf
and some different examples here: http://www.texample.net/tikz/examples/
I use ipe for all my figures. Snapping, vector graphics, and the ability to include latex are some of the standout features.