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Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems. Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain geometric results. According to Felix Klein
Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates.
Geometry as presented by Euclid in the ''Elements'' is the quintessential example of the use of the synthetic method. It was the favoured method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry and non-Euclidean geometries. For example the geometer Jakob Steiner (1796 – 1863) hated analytic geometry, and always gave preference to synthetic methods.

Logical synthesis

The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives: * Primitives are the most basic ideas. Typically they include both objects and relationships. In geometry, the objects are things such as ''points'', ''lines'' and ''planes'', while a fundamental relationship is that of ''incidence'' – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs, the point being that the primitive terms are just empty placeholders and have no intrinsic properties. * Axioms are statements about these primitives; for example, ''any two points are together incident with just one line'' (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the ''building blocks'' of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a theorem.

Properties of axiom sets

There is no fixed axiom set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are also examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular. Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and "betweeness" are also optional, for example, discrete geometries may be created by discarding or modifying them. Following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development.

History

Proofs using synthetic geometry

Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles. Examples of such proofs can be found in the articles Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem, Law of cosines, and others that are linked to here.

Computational synthetic geometry

In conjunction with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory.

* Foundations of geometry * Incidence geometry * Synthetic differential geometry

Notes

References

* * * Halsted, G. B. (1896
Elementary Synthetic Geometry
via Internet Archive * Halsted, George Bruce (1906
Synthetic Projective Geometry
via Internet Archive. * Hilbert & Cohn-Vossen, ''Geometry and the imagination''. * * {{Authority control Category:Fields of geometry