# Synthetic Geometry

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

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