--- title: "FindGeometricConjectures" language: "en" type: "Symbol" summary: "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." keywords: - geometric conjecture canonical_url: "https://reference.wolfram.com/language/ref/FindGeometricConjectures.html" source: "Wolfram Language Documentation" related_guides: - title: "Plane Geometry" link: "https://reference.wolfram.com/language/guide/PlaneGeometry.en.md" - title: "Synthetic Geometry" link: "https://reference.wolfram.com/language/guide/SyntheticGeometry.en.md" - title: "Geometric Computation" link: "https://reference.wolfram.com/language/guide/GeometricComputation.en.md" - title: "Solvers over Regions" link: "https://reference.wolfram.com/language/guide/GeometricSolvers.en.md" related_functions: - title: "RandomInstance" link: "https://reference.wolfram.com/language/ref/RandomInstance.en.md" - title: "GeometricScene" link: "https://reference.wolfram.com/language/ref/GeometricScene.en.md" - title: "GeometricAssertion" link: "https://reference.wolfram.com/language/ref/GeometricAssertion.en.md" - title: "Reduce" link: "https://reference.wolfram.com/language/ref/Reduce.en.md" - title: "FindEquationalProof" link: "https://reference.wolfram.com/language/ref/FindEquationalProof.en.md" related_tutorials: - title: "Synthetic Geometry" link: "https://reference.wolfram.com/language/tutorial/SyntheticGeometry.en.md" --- [EXPERIMENTAL] # 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`](https://reference.wolfram.com/language/ref/Method.en.md) [`Automatic`](https://reference.wolfram.com/language/ref/Automatic.en.md) method to use * Random seeds may be given using ``Method -> {"RandomInstance", RandomSeeding -> seed}``. ## Examples (5) ### Basic Examples (1) Represent a scene with a circumscribed triangle with the diameter as an edge: ```wl In[1]:= RandomInstance[GeometricScene[{a, b, c, o}, {Triangle[{a, b, c}], CircleThrough[{a, b, c}, o], o == Midpoint[{a, c}]}]] Out[1]= GeometricScene[{{a -> {1.4305253895316499, 3.067736334828689}, b -> {-3.1025813969730245, 3.7295272003897817}, c -> {-3.66574764734203, -0.1280240702491874}, o -> {-1.11761112890519, 1.4698561322897508}}, {C["GeometricQuantity"][C["Unspecified"]["radius", "{{C[\"GeometricPoint\"][a], C[\"GeometricPoint\"][b], C[\"GeometricPoint\"][c]}, {1, 2, 0}}"]\ ] -> 3.0076936313480007}}, {Triangle[{a, b, c}], CircleThrough[{a, b, c}, o], o == Midpoint[{a, c}]}, {}] ``` Find conjectures that appear to hold for the scene: ```wl In[2]:= FindGeometricConjectures[%] Out[2]= GeometricScene[{{a -> {1.4305253895316499, 3.067736334828689}, b -> {-3.1025813969730245, 3.7295272003897817}, c -> {-3.66574764734203, -0.1280240702491874}, o -> {-1.11761112890519, 1.4698561322897508}}, {C["GeometricQuantity"][C["Unspe ... "{{C[\"GeometricPoint\"][a], C[\"GeometricPoint\"][b], C[\"GeometricPoint\"][c]}, {1, 2, 0}}"]\ ] -> 3.0076936313480007}}, {Triangle[{a, b, c}], CircleThrough[{a, b, c}, o], o == Midpoint[{a, c}]}, {PlanarAngle[{a, b, c}] == 90*Degree}] ``` Discover Thales's theorem: ```wl In[3]:= FindGeometricConjectures[%, PlanarAngle[{__}] == 90°]["Conclusions"] Out[3]= {Inactive[PlanarAngle][{a, b, c}] == 90 °} ``` ### Scope (2) Find two instances of a scene with two sets of collinear points: ```wl In[1]:= 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] Out[1]= {GeometricScene[{{a -> {3.3978465813297087, -10.36150510417867}, b -> {-2.2238636756687415, -8.988501287288305}, c -> {-6.92921346106794, -7.839302392330297}, d -> {5.0333258153111835, 1.9604888367247082}, e -> {0.9379744247677756, 6.411313 ... 9828075399239168, -4.611510943036213}}, {}}, {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}], RGBColor[1, 0, 0]]}, {}]} ``` Discover Pappus's hexagon theorem by searching for conjectures satisfied by both instances: ```wl In[2]:= FindGeometricConjectures[%]["Conclusions"] Out[2]= {Inactive[GeometricAssertion][{z, y, x}, "Collinear"], Inactive[PlanarAngle][{c, y, f}] == Inactive[PlanarAngle][{d, y, a}], Inactive[PlanarAngle][{b, z, e}] == Inactive[PlanarAngle][{f, z, c}], Inactive[PlanarAngle][{b, x, e}] == Inactive[PlanarAngle][{d, x, a}]} ``` --- Represent a scene with a pentagram where some angles are known: ```wl In[1]:= 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] Out[1]= GeometricScene[{{a -> {-8.068821671106983, -0.6894914107748263}, b -> {-1.878804120368458, 0.9184060359145161}, c -> {1.9051635402252574, 1.9013178323684423}, d -> {10.6093861345504, 4.16229713527108}, e -> {3.5629207132697216, -1.726030728 ... Line[{g, h, b, i}], Line[{i, c, e, j}], Line[{j, f, h, a}], PlanarAngle[{b, c, e}, "Counterclockwise"] == 100*Degree, PlanarAngle[{f, h, b}, "Counterclockwise"] == 110*Degree, PlanarAngle[{h, a, b}, "Counterclockwise"] == 35*Degree}, {}] ``` Solve for missing angles: ```wl In[2]:= FindGeometricConjectures[%, PlanarAngle[{__}] == _ ? NumericQ][ "Conclusions"] Out[2]= {Inactive[PlanarAngle][{f, j, e}] == 45 °, Inactive[PlanarAngle][{b, i, c}] == 25 °} ``` Only return one missing angle: ```wl In[3]:= FindGeometricConjectures[%%, PlanarAngle[{__}] == _ ? NumericQ, 1][ "Conclusions"] Out[3]= {Inactive[PlanarAngle][{f, j, e}] == 45 °} ``` ### Applications (2) Iteratively take circumcenters: ```wl In[1]:= 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] Out[1]= GeometricScene[{{a -> {-0.15215527166430298, -3.6122037657904373}, b -> {-4.707002146239533, 3.1480435540513616}, c -> {4.158147342176919, 2.999413512835222}, k -> {-0.17193115286984678, 0.7181727089902079}, o -> {-0.30586154049192477, 1.19 ... ngleCenter[{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}]}, {}] ``` Discover Kosnita's theorem: ```wl In[2]:= FindGeometricConjectures[%]["Conclusions"] Out[2]= {Inactive[GeometricAssertion][{oc, k, c}, "Collinear"], Inactive[PlanarAngle][{b, k, a}] == Inactive[PlanarAngle][{ob, k, oa}], Inactive[PlanarAngle][{b, c, o}] == Inactive[PlanarAngle][{c, b, o}], Inactive[PlanarAngle][{a, c, o}] == Inactive[Plana ... arAngle][{b, a, o}], EuclideanDistance[o, a] == EuclideanDistance[b, o] == EuclideanDistance[o, c], Inactive[Area][Polygon[{a, b, c}]] == Inactive[Area][Polygon[{a, b, o}]] + Inactive[Area][Polygon[{a, o, c}]] + Inactive[Area][Polygon[{o, b, c}]]} ``` --- Describe a scene with two squares and a quadrilateral formed by taking midpoints: ```wl In[1]:= 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] Out[1]= GeometricScene[{{a -> {-2.74904252759296, 0.16353074082065616}, b -> {-0.07300936153741817, -0.965181969441688}, bb -> {-1.7338525027166778, -3.236962925296686}, c -> {1.05570334865533, 1.7108511966030437}, cc -> {1.6666411633310672, -2 ... {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], RGBColor[1, 0, 0]]}, {}] ``` Discover the Finsler–Hadwiger theorem: ```wl In[2]:= FindGeometricConjectures[%, GeometricAssertion[_, "Regular"]]["Conclusions"] Out[2]= {Inactive[GeometricAssertion][Polygon[{q, r, s, t}], "Regular"]} ``` ## See Also * [`RandomInstance`](https://reference.wolfram.com/language/ref/RandomInstance.en.md) * [`GeometricScene`](https://reference.wolfram.com/language/ref/GeometricScene.en.md) * [`GeometricAssertion`](https://reference.wolfram.com/language/ref/GeometricAssertion.en.md) * [`Reduce`](https://reference.wolfram.com/language/ref/Reduce.en.md) * [`FindEquationalProof`](https://reference.wolfram.com/language/ref/FindEquationalProof.en.md) ## Tech Notes * [Synthetic Geometry](https://reference.wolfram.com/language/tutorial/SyntheticGeometry.en.md) ## Related Guides * [Plane Geometry](https://reference.wolfram.com/language/guide/PlaneGeometry.en.md) * [Synthetic Geometry](https://reference.wolfram.com/language/guide/SyntheticGeometry.en.md) * [Geometric Computation](https://reference.wolfram.com/language/guide/GeometricComputation.en.md) * [Solvers over Regions](https://reference.wolfram.com/language/guide/GeometricSolvers.en.md) ## History * [Introduced in 2019 (12.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn120.en.md)