Without using formulas means: based on a system of axioms and theorems.

"The first systematic approach for synthetic geometry is Euclid's Elements."

Examples are Thales' theorem, the intercept theorem, and similar.

Coordinates, the opposite of synthetic geometry weren't used before Descartes in the 17th century.

The Wikipedia page of "synthetic geometry" is available in 17 languages. They are usually of different content, i.e. not simple translations of each other. Maybe you can read another one, or find even your native language among them. It is always an advantage to look up the content of other languages. E.g. the German Wikipedia pages are often less theoretical and contain more proofs than the English ones.