I 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/c

I 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/apresentacaoDaysInLogic06.pdf http://www.mat.uc.pt/~pedro/cientificos/Publicacoes/apresent... and http://argo.matf.bg.ac.rs/publications/2011/area.pdf 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.