I found the introduction to automated theorem proving using the Ritt-Wu method of characteristic sets in [Cox, Little, and O'Shea](https://www.springer.com/us/book/9781441922571) to be quite enlightening for pedagogy on introductory algebraic geometry.

I don't know if any such theorem proven by the Ritt-Wu method "surprising" but it was a great way to tie a lot of the ideas in CLO's previous chapters.

[Chou and Gao](http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=7E0EF1B903942A126EB0D7B0BC53A232?doi=10.1.1.132.741&rep=rep1&type=pdf) describe proving Pascal's theorem, the Butterfly theorem, Morley's theorem, etc. using the Ritt-Wu method.