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 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.

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Hilbert's axioms concept
Synthetic geometry definition

Tag	`YH5PGKNZKD`
URI	`m2://YH5PGKNZKD`
Kind	definition
Writing	W0 stub
MSC	51-01, 51M05, 51A05
Defines	Synthetic geometry
Uses	Hilbert's axioms

Synthetic geometry

Definition

Remarks

Reading path

Further reading

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