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Synthetic Geometry—Wolfram Language Documentation
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TECH NOTE 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. Scene Representation and Visualization Synthetic geometry scenes are represented in the Wolfram Language with GeometricScene. Scenes contain parameters representing named points and quantities, as well as hypotheses consisting of symbolic 2D regions and assertions involving those parameters. Conclusions can also be added to represent geometric theorems or conjectures that those hypotheses entail. Particular instances of a scene can be represented by specifying coordinates and values for all of the parameters appearing in the scene. Multiple instances of the same scene can be represented with a single GeometricScene object. GeometricScene[{p1,p2,…},{hyp1,hyp2,…}] abstract 2D geometric scene defined by the hypotheses hypi in terms of the symbolic points pi GeometricScene[{{p1,p2,…},{k1,k2,…}},hyps] scene whose hypotheses depend on the symbolic scalar values ki GeometricScene[{p1{x1,y1},p2{x2,y2},…},hyps] specific instance of a geometric scene with explicit coordinates for each point GeometricScene[{{p1{x11,y11},…},{p1{x21,y21},…},…},hyps] collection of specific instances of a scene GeometricScene[…,…,{con1,con2,…}] scene together with some conclusions coni about it GeometricScene[{scene1,scene2,…}] combines several scene instances into one scene object Wrapper for geometric scenes. The simplest geometric scenes consist of only a list of points and a list of hypotheses. Here is a scene consisting of a circumscribed triangle with one side on the diameter: Geometric scenes can be visualized by finding a particular instance of the scene using RandomInstance. Function to find instances of a geometric scene. Display an interactive instance of the scene: This finds several instances of the scene: Many 2D regions can be used as hypotheses in a geometric scene. All regions appearing in the hypotheses of a geometric scene are assumed to be nondegenerate and are displayed in instances of that scene: AngleBisector[{p,q,r}] infinite line bisecting the angle ∠ p q r Circle[p,r] circle of radius r centered at the point p CircleThrough[{p1,p2,…}] circle passing through the points pi Circumsphere[{p1,p2,p3}] sphere passing through the points pi Disk[p,r] filled disk of radius r centered at the point p HalfLine[{p,q}] half-infinite line, or ray, starting at the point p and passing through the point q InfiniteLine[{p,q}] infinite line passing through the points p and q Insphere[{p1,p2,p3}] sphere tangent to the sides of the triangle △ p1 p2 p3 Line[{p1,p2,…}] line segment passing through the points pi, in that order Midpoint[{p,q}] midpoint of the line segment p q PerpendicularBisector[{p,q}] perpendicular bisector of the line segment p q Point[p] point p Polygon[{p1,p2,…}] polygon with vertices pi RegionBoundary[reg] boundary of the region reg RegionCentroid[reg] centroid of the region reg RegionNearest[reg,p] point closest to the point p in the region reg Triangle[{p,q,r}] triangle with vertices p, q and r TriangleCenter[{p,q,r},spec] center of triangle △ p q r specified by spec TriangleConstruct[{p,q,r},spec] constructed geometric region defined by the triangle △ p q r specified by spec This is a more complex geometric scene: Hypotheses can also be geometric assertions or equations involving geometric quantities defined on elements of the scene. p∈reg assertion that the point p is an element of the region reg x1… assertion that the regions/quantities xi are equal GeometricAssertion[objs,prop] assertion that the objects objs satisfy the property prop GeometricStep[hyps,label] step consisting of multiple hypotheses RegionMember[reg,p] assertion that the point p is a member of the region reg This is the scene description for Brahmagupta's theorem: GeometricScene also supports style: Find a perfect polygon dissection: Construct an equilateral triangle: The Wolfram Language can use these scene descriptions to find conjectures that may hold for a geometric scene. Function to find conjectures for a geometric scene. Discover Thales's theorem: Discover Pappus's hexagon theorem: Discover Kosnita's theorem: Discover the Finsler–Hadwiger theorem: The Wolfram Language can perform logical reasoning on geometric scenes using such functions as GeometricSolveValues. GeometricSolveValues[scene,expr] solve for the symbolic geometric quantity expr defined by scene GeometricSolveValues[scene,{expr1,expr2,…}] solve for multiple quanitites expr1,expr2, … defined by scene Function to solve for values in a geometric scene. Find the area of the quadrilateral in the following scene: Find the area of a triangle in both a general scene and a specific instance of that scene: Reasoning on geometric objects can be done using GeometricTest. GeometricTest[obj,prop] test whether the geometric object obj satisfies prop GeometricTest[{obj1,obj2,…},prop] test whether the obji satisfy prop GeometricTest[objs,prop1,prop2,…] test whether objs satisfy each of the propi Function to test whether geometric objects satisfy a given property. Determine if three points are collinear: Find conditions for three abstract points to be collinear: Find conditions for multiple predicates to hold for a single geometric object:
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