I found the introduction to automated theorem proving using the Ritt-Wu method of characteristic sets in Cox, Little, and O'Shea 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 describe proving Pascal's theorem, the Butterfly theorem, Morley's theorem, etc. using the Ritt-Wu method.