The Wolfram Language provides not only extensive support for analytic geometry, but also support for the symbolic representation of synthetic geometry scenes in a form suitable for automated coordinate-independent reasoning, as well as automated visualization.

GeometricScene — symbolic representation of a geometric scene defined by constraints

RandomInstance — find and display a random instance of a geometric scene

GeometricSolveValues — solve for values in a geometric scene

GeometricTest — get conditions for a geometric assertion outside of a geometric scene

FindGeometricConjectures — find conjectures about a geometric scene

GeometricScene — entities representing notable theorems and constructions

Assertions in Geometric Scenes

GeometricAssertion — assert geometric properties or relations (convex, parallel, congruent, ...)

Equal (  ) — assert equality of geometric objects or quantities

Element ( ∈ ), RegionMember — assert that a point is in a geometric region

UnconstrainedParameters — option defining unconstrained points or quantities in a scene

GeometricStep — group hypotheses into a step in the construction of a geometric scene

Basic Geometric Objects

Line — straight line through symbolically specified points

HalfLine, InfiniteLine — half-infinite and doubly infinite lines through pairs of points

Triangle, Polygon — triangle, polygon defined by corners

Circle, Disk — circle, disk defined by center and radius

RegularPolygon — regular polygon with sides, defined by center, etc.

Triangle Functions

TriangleCenter — orthocenter, circumcenter, incenter, ... of triangles

TriangleConstruct — incircle, circumcircle, altitude, ... of triangles

TriangleMeasurement — circumradius, height, ... of triangles

Objects Created by Geometric Constructions

CircleThrough — circle through specified points

Midpoint — midpoint of a line

PerpendicularBisector — infinite perpendicular at the midpoint of a line

AngleBisector — infinite line bisecting an angle

RegionCentroid — centroid of any region

RegionBoundary — boundary of any region

RegionNearest — point in a region nearest to a given point

Measurements on Geometric Objects

EuclideanDistance — Euclidean distance between points

PlanarAngle — planar angle defined by three points

PolygonAngle — vertex angle of a polygon

Area — area of a geometric region

Perimeter — perimeter of a geometric region

ArcLength — arc lengths of circles, etc.

RegionMeasure — measure of a 1D, 2D, etc. region

RegionDistance — shortest distance from a point to a geometric region

SignedRegionDistance — distance positive/negative for points outside/inside