Depends on your definition of recent. Formal proofs and logic have existed for a long time by their rigorisation only really began in the 19th century and a fully-developed theory of mathematics...
The phrase 'mathematical logic' means any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity
This cannot be done using construction techniques of classic geometry. It is possible in non-Euclidean geometries, but is essentially a different problem than the one we think of in Euclidean...