Earlier quoted context omitted.
I think this illustrates one of the preconceptions this challenge is meant to work against. Namely, people think of coordinate bashing as "computational" and purely synthetic/geometric solutions as "insightful". But there's no reason that a computer would find the second way more difficult. After all, synthetic geometry is just another system of rules, like ordinary algebra. Us humans might fall back on coordinate ba…
The current situation doesn't reflect that though. We have a algorithm (Gröbner bases) for doing coordinate bashes which is guaranteed to succeed, whereas as far as I know we don't have an analogous algorithm in terms of synthetic geometry (other than the direct translation of coordinate bashing).
Re: IMO Grand Challenge
#71I think there is actually a guaranteed synthetic-geometry technique, where the main idea is to compare areas of regions. See e.g. http://www.mat.uc.pt/~pedro/cientificos/Publicacoes/apresent... and http://argo.matf.bg.ac.rs/publications/2011/area.pdf. I vaguely remember hearing many years ago that the Chinese IMO team had been trained in applying something like this method, but I've no idea whether it was true then or whether anything like it is true now.