---
tag: YH5PGKNZKD
kind: definition
name: Synthetic geometry
msc: [51-01, 51M05, 51A05]
defines: VY6ZCZY8DA
uses: [KM1FB8XZ2R]
sources:
  - url: https://en.wikipedia.org/wiki/Synthetic_geometry
---

## Definition

*Synthetic geometry* is the study of geometry from axioms whose primitive notions are geometric objects and relations, such as points, lines, incidence, betweenness and congruence.
Its proofs derive theorems from the axioms directly, without coordinates in a field or a ring.

## Remarks

Example: [[Hilbert's axioms|KM1FB8XZ2R]] give a synthetic geometry, and a proof of the congruence criterion side-side-side from them is a synthetic proof.
Non-example: write the vertices of a parallelogram as vectors $0$, $u$, $v$, $u + v$, and check that both diagonals have midpoint $(u + v)/2$.
This proof that the diagonals bisect each other is analytic, not synthetic.
Coordinates may come back as a result: for example, Desargues's theorem in a projective plane allows one to build a division ring of coordinates from the axioms.