[generationP]

Ah, thanks! It seems to be this one http://www.mmrc.iss.ac.cn/~xgao/paper/jar-gdbase.pdf plus some algebra for adding angles and lengths. As I hoped, they are using oriented angles mod 180° ("full-angles") to get rid of betweenness conditions. They are also using evaluations at rational (I think) coordinates to check that nondegenerateness conditions are satisfied when the data is in general position. This is not as nice as I had hoped for, but certainly no worse than what contestants do at the IMO. And because a bunch of methods are off-limits (congruence criteria, probably anything involving undirected unsquared lengths), I'm not surprised that their proofs are longer/slower than human ones -- they're better proofs.