FindGeometricConjectures
FindGeometricConjectures[scene]
finds conjectures that appear to hold for the GeometricScene object scene and adds these conjectures to the scene object.
FindGeometricConjectures[{scene1,scene2,…}]
finds conjectures that appear to hold for all instances scenei of a geometric scene and returns a combined scene with the conjectures added.
FindGeometricConjectures[scenes,patt]
adds only conjectures that match the pattern patt.
FindGeometricConjectures[scenes,patt,n]
adds only up to n conjectures.
Details and Options
- If scene is a GeometricScene object with one or more instances, then FindGeometricConjectures returns a GeometricScene with added conclusions that are conjectured to hold for each instance of the scene.
- The conclusions of a GeometricScene object can be obtained from GeometricScene[…]["Conclusions"].
- If no scene instances are given, RandomInstance is used to generate instances from which to gather conjectures.
- FindGeometricConjectures[{scene1,scene2,…}] is equivalent to FindGeometricConjectures[GeometricScene[{scene1,scene2,…}]].
- Possible values of patt include GeometricAssertion[_,"Perpendicular"], _==Midpoint[{_,_}], ….
- The scenei must have the same list of points, quantities and hypotheses, but may represent different instances of the same scene.
- The following options can be given:
-
Method Automatic method to use - Random seeds may be given using Method{"RandomInstance",RandomSeedingseed}.
Examples
open all close allBasic Examples (1)
Represent a scene with a circumscribed triangle with the diameter as an edge:
RandomInstance[GeometricScene[{a, b, c, o}, {Triangle[{a, b, c}], CircleThrough[{a, b, c}, o], o == Midpoint[{a, c}]}]]Find conjectures that appear to hold for the scene:
FindGeometricConjectures[%]FindGeometricConjectures[%, PlanarAngle[{__}] == 90°]["Conclusions"]Scope (2)
Find two instances of a scene with two sets of collinear points:
RandomInstance[GeometricScene[{a, b, c, d, e, f, x, y, z}, {Line[{{a, b, c}}], Line[{d, e, f}], Line[{a, x, e}], Line[{b, x, d}], Line[{a, y, f}], Line[{c, y, d}], Line[{b, z, f}], Line[{c, z, e}], Style[InfiniteLine[{x, z}], Red]}], 2, RandomSeeding -> 1]Discover Pappus's hexagon theorem by searching for conjectures satisfied by both instances:
FindGeometricConjectures[%]["Conclusions"]Represent a scene with a pentagram where some angles are known:
RandomInstance[GeometricScene[{a, b, c, d, e, f, g, h, i, j}, {Line[{a, b, c, d}], Line[{d, e, f, g}], Line[{g, h, b, i}], Line[{i, c, e, j}], Line[{j, f, h, a}], PlanarAngle[{b, c, e}, "Counterclockwise"] == 100°, PlanarAngle[{f, h, b}, "Counterclockwise"] == 110°, PlanarAngle[{h, a, b}, "Counterclockwise"] == 35°}], RandomSeeding -> 1234]FindGeometricConjectures[%, PlanarAngle[{__}] == _ ? NumericQ][
"Conclusions"]Only return one missing angle:
FindGeometricConjectures[%%, PlanarAngle[{__}] == _ ? NumericQ, 1][
"Conclusions"]Applications (2)
Iteratively take circumcenters:
RandomInstance[GeometricScene[{a, b, c, o, oa, ob, oc, k}, {o == TriangleCenter[{a, b, c}, "Circumcenter"], oa == TriangleCenter[{o, b, c}, "Circumcenter"], ob == TriangleCenter[{a, o, c}, "Circumcenter"], oc == TriangleCenter[{a, b, o}, "Circumcenter"], Line[{a, k, oa}], Line[{b, k, ob}], Line[{c, oc}]}], RandomSeeding -> 17]FindGeometricConjectures[%]["Conclusions"]Describe a scene with two squares and a quadrilateral formed by taking midpoints:
RandomInstance[GeometricScene[{a, b, c, d, bb, cc, dd, q, r, s, t}, {GeometricAssertion[{Polygon[{a, b, c, d}], Polygon[{a, bb, cc, dd}]}, "Regular", "Counterclockwise"], q == Midpoint[{bb, d}], r == Midpoint[{a, c}], s == Midpoint[{b, dd}], t == Midpoint[{a, cc}], Style[Polygon[{q, r, s, t}], Opacity[0.4], Red]}], RandomSeeding -> 1]Discover the Finsler–Hadwiger theorem:
FindGeometricConjectures[%, GeometricAssertion[_, "Regular"]]["Conclusions"]Text
Wolfram Research (2019), FindGeometricConjectures, Wolfram Language function, https://reference.wolfram.com/language/ref/FindGeometricConjectures.html.
CMS
Wolfram Language. 2019. "FindGeometricConjectures." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FindGeometricConjectures.html.
APA
Wolfram Language. (2019). FindGeometricConjectures. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindGeometricConjectures.html