---
tag: VY6ZCZY8DA
kind: concept
name: Synthetic geometry
msc: [51-01, 51M05, 51A05]
sources:
  - url: https://en.wikipedia.org/wiki/Synthetic_geometry
same_as:
  wikidata: Q249148
  wikipedia: Synthetic geometry
---

## Summary

Synthetic geometry is geometry done from axioms about figures, without coordinates or formulas.
Its proofs reason directly with points, lines, circles and the relations between them, as in [[Euclid's Elements|83WHDAV68J]].
It is opposite to analytic geometry, which translates figures into equations.
Axiom systems such as [[Hilbert's axioms|KM1FB8XZ2R]] and the theory of [[projective planes|DANFNN8C4Y]] are modern forms of synthetic geometry.

## Explanations

### L1 human

There are two ways to prove that the medians of a triangle meet in one point.
One can give the vertices coordinates and solve equations; that is analytic geometry.
Or one can argue with the figure itself, using similar triangles; that is synthetic geometry.
Synthetic proofs often show why a result is true, while analytic proofs are often more mechanical.