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Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems.

The geometry of Euclid was indeed synthetic, though not all of the books covered topics of pure geometry. The heyday of synthetic geometry can be considered to have been the nineteenth century; when methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a synthetic development of projective geometry.

For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. The close axiomatic study of Euclidean geometryEuclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry someti led to the discovery of non-Euclidean geometryThe term non-Euclidean geometry (also spelled: non-Euclidian geometry describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of par. The question is whether this is success or failure.

If the axiom set is not categorical (so that there is more than one model) one has the geometry/geometries debate to settle. That's not a serious issue for a modern axiomatic mathematician, since the implication of axiomFor the algebra software named Axiom, see Axiom (algebra software). For the 1970s Australian rock music group, see Axiom (band). In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up is now starting point for theory rather than self-evident plank in platform based on intuition. And since the Erlangen programme of Klein the geometrical nature of a geometry has been seen as the connection of symmetrySymmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. The three main symmetri and the content of propositions, rather than the style of development.

In relation with computational geometryIn computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and the study of such problems is also con, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory.