Let's see. Most of the results of projective geometry are much harder to show by more elementary methods. I think the reason for this is that they rest more on the incidence axioms while the elementary methods play more with the metric properties.

An interesting question that arises for any of these examples is to detect why is that the case that the elementary proves are harder.

A classical example is the constructibility of regular polyhedral with ruler and compass.

Let's see. Most of the results of projective geometry are much harder to show by more elementary methods. I think the reason for this is that they rest more on the incidence axioms while the elementary methods play more with the metric properties.

An interesting question that arises for any of these examples is to detect why is that the case that the elementary proofs are harder.

A classical example is the constructibility of regular polyhedral with ruler and compass.