# GeometricAssertion

GeometricAssertion[obj,prop]

represents the assertion that the geometric object obj satisfies prop.

GeometricAssertion[{obj1,obj2,},prop]

represents the assertion that the obji satisfy prop.

GeometricAssertion[objs,prop1,prop2,]

represents the assertion that objs satisfies each of the propi.

# Details   • GeometricAssertion can be used in stating hypotheses in GeometricScene.
• The propi can either be predicates that apply to individual geometric objects or relations that hold between geometric objects.
• The following predicates for polygons can be used:
•  "Clockwise" the vertices are in clockwise order "Convex" the polygon is convex "Counterclockwise" the vertices are in counterclockwise order "Cyclic" the vertices lie on a circle "Equiangular" the interior angles are all equal "Equilateral" the side lengths are all equal "Regular" the polygon is regular
• The following relations between points can be used:
•  "Clockwise" the points are in clockwise order if successively joined "Collinear" the points lie on a line "Counterclockwise" the points are in counterclockwise order if successively joined "CyclicallyOrdered" the points are in either clockwise or counterclockwise order "Distinct" the points are distinct {"EqualAngles",pt} successive angles subtended at pt are equal {"OppositeSides", } the points are on opposite sides of line {"SameSide", line} the points are on the same side of line
• The following relations between lines can be used:
•  "Antiparallel" the lines are parallel but in opposite directions "Concurrent" the lines intersect at a common point {"Concurrent",pt} the lines intersect at the point pt "MatchingParallel" the lines are parallel and in the same direction "Nondegenerate" the lines lie on distinct infinite lines "Parallel" the lines are parallel (with any relative orientation) "Perpendicular" the lines are perpendicular
• The following relations between geometric objects can be used:
•  "Congruent" the objects are congruent {"Inside",circle} the objects are inside the circle {"Outside",circle} the objects are outside the circle "PairwiseTangent" the objects are tangent to each other, in pairs "Similar" the objects are similar "Tangent" the objects are all tangent to each other at a common point {"Tangent",pt} the objects are all tangent at the point pt
• If prop is a predicate, then GeometricAssertion[{obj1,obj2,},prop] is equivalent to GeometricAssertion[obj1,prop]&&GeometricAssertion[obj2,prop]&&.
• GeometricAssertion[objs,prop1,prop2,] is equivalent to GeometricAssertion[objs,prop1]&&GeometricAssertion[objs,prop2]&&.

# Examples

open allclose all

## Scope(24)

### Predicates(8)

Describe a scene with a clockwise-oriented polygon:

Describe a scene with a convex polygon:

Describe a scene with a counterclockwise-oriented polygon:

Describe a scene with a cyclic polygon:

Describe a scene with an equiangular polygon:

Describe a scene with an equilateral polygon:

Describe a scene with a regular polygon:

Describe a scene with a simple polygon:

### Relations(16)

#### Relations between Points(6)

Describe a scene with a clockwise-oriented polygon:

Describe a scene with collinear points:

Describe a scene with a counterclockwise-oriented polygon:

Describe a scene with four points cyclically ordered:

Describe a scene where two points are asserted to be distinct:

Describe a scene with a trisected angle:

#### Relations between regions(10)

Describe a scene with two antiparallel lines:

Describe a scene with concurrent lines:

Specify a point of concurrency:

Describe a scene with congruent triangles:

Describe a scene with two matching parallel lines:

Describe a scene with pairwise tangent circles:

Describe a scene with parallel lines:

Describe a scene with two perpendicular lines:

Describe a scene with similar polygons:

Describe a scene with a tangent line to a circle:

Describe a scene with mutually tangent circles:

Introduced in 2019
(12.0)
|
Updated in 2020
(12.1)