Euclid Book 4
Proposition 2
Construction
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Euclid Book 4 Proposition 3Statement
To inscribe inside a given circle a triangle similar to a given triangle.
Statement HTML
To inscribe inside a given circle a triangle similar to a given triangle.
Construction steps
1. Let the circle through the point A centered at O be the given circle and △DEF the given triangle.
2. Construct that is tangent to the circle at point A.
GH
3. Find a point C on the circle such that ∠CAH∠DEF.
4. Find a point B on the circle such that ∠BAG∠DEF.
5. Join . △ABC has been inscribed in the given circle and similar to the given △DEF.
BC
Construction Steps HTML
Original statement
ϵἰς τὸν δοθέντα κύκλον τῷ δοθέντι τριγώνῳ ἰσογώνιον τρίγωνον ἐγγράψαι.
English translation
In a given circle to inscribe a triangle equiangular with a given triangle.
Computable version
GeometricScene[{{A.,O.,B.,C.,D.,E.,F.,G.,H.},{}},{GeometricStep[{CircleThrough[{A.},O.],Triangle[{D.,E.,F.}]}],GeometricStep[{GeometricAssertion[{Line[{G.,A.,H.}],CircleThrough[{A.},O.]},"Tangent"]}],GeometricStep[{C.∈CircleThrough[{A.},O.],Line[{A.,C.}],PlanarAngle[{C.,A.,H.}]PlanarAngle[{D.,E.,F.}]}],GeometricStep[{B.∈CircleThrough[{A.},O.],Line[{A.,B.}],PlanarAngle[{B.,A.,G.}]PlanarAngle[{D.,F.,E.}]}],GeometricStep[{Line[{B.,C.}]}]},{GeometricAssertion[{Triangle[{A.,B.,C.}],Triangle[{D.,E.,F.}]},"Similar"]}]//RandomInstance
Out[]=
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PlaneGeometry`GeometrySceneDrawerDump`makeGraph icsFromRulesAbsoluteThickness[2], EdgeForm Thickness[Large],Opacity[0.2], Opacity[0.50000000],EdgeForm Thickness[Large], Polygon[{ GeometricPoint GeometricPoint GeometricPoint Opacity[0.50000000],EdgeForm Thickness[Large], Polygon[{ GeometricPoint GeometricPoint GeometricPoint trySceneDrawerDump`makeAngleGraphics[{Plan eGeometry`GeometrySceneDrawerDump`geoAn gle[ GeometricPoint GeometricPoint GeometricPoint Polygon[{ GeometricPoint GeometricPoint GeometricPoint ["GeometricPoint"][D.], C["GeometricPoint"][E.], C["GeometricPoint"][F.]}]],PlaneGeometry` GeometrySceneDrawerDump`makeAngleGraphics[{PlaneGeometry`GeometrySceneDrawerDump`geoAngle[ GeometricPoint GeometricPoint GeometricPoint Polygon[{ GeometricPoint GeometricPoint GeometricPoint ["GeometricPoint"][D.], C["GeometricPoint"][E.], C["GeometricPoint"][F.]}]],PlaneGeometry` GeometrySceneDrawerDump`makeAngleGraphics[{PlaneGeometry`GeometrySceneDrawerDump`geoAngle[ GeometricPoint GeometricPoint GeometricPoint Polygon[{ GeometricPoint GeometricPoint GeometricPoint ["GeometricPoint"][D.], C["GeometricPoint"][F.], C["GeometricPoint"][E.]}]],PlaneGeometry` GeometrySceneDrawerDump`makeAngleGraphics[{PlaneGeometry`GeometrySceneDrawerDump`geoAngle[ GeometricPoint GeometricPoint GeometricPoint Polygon[{ GeometricPoint GeometricPoint GeometricPoint ["GeometricPoint"][D.], C["GeometricPoint"][F.], C["GeometricPoint"][E.]}]],{Dashing[{Small, Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint Small}],CapForm[None], Line[{ GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["Geometri cPoint"][C["Unspecified"]["ref", "{1, 1, 0}", "1"]], C["Geometric Point"][C["Unspecified"]["ref", "{1, 1, 0}", "2"]]}, {1, 1, 0}}],Circle GeometricPoint GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["Geometri cPoint"][C["Unspecified"]["ref", "{2, 1, 1, 2, 0}", "1"]], C["Geo metricPoint"][C["Unspecified"]["r ef", "{2, 1, 1, 2, 0}", "2"]]}, {2, 1, 1, 2, 0}}],Circle GeometricPoint GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["Geometri cPoint"][C["Unspecified"]["ref", "{3, 1, 2, 0}", "1"]], C["Geomet ricPoint"][C["Unspecified"]["ref", "{3, 1, 2, 0}", "2"]]}, {3, 1, 2, 0}}],Circle GeometricPoint GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["Geometri cPoint"][C["Unspecified"]["ref", "{4, 1, 2, 0}", "1"]], C["Geomet ricPoint"][C["Unspecified"]["ref", "{4, 1, 2, 0}", "2"]]}, {4, 1, 2, 0}}], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0],Point[ GeometricPoint ,AbsolutePointSize[17], Point[ GeometricPoint GeometricPoint Opacity[0], Point[ GeometricPoint GeometricPoint 1.41653}, GeometricPoint 3.83752}, GeometricPoint -0.59544}, GeometricPoint 2.01755}, GeometricPoint -1.64007}, GeometricPoint 1.64007}, GeometricPoint 4.90585}, GeometricPoint -2.07279}, GeometricPoint -2.41709}, GeometricPoint Unspecified tangent,{Line[{C["GeometricPoint"][G.], C["GeometricPoint"][A.], C["GeometricPoint"][H.]}], geoCircleThr ough[{C["GeometricPoint"][A.], C["Geome tricPoint"][C["Unspecified"]["ref", "{2, 1, 1, 2, 0}", "1"]], C["Geom etricPoint"][C["Unspecified"]["ref ", "{2, 1, 1, 2, 0}", "2"]]}, C["GeometricPoint"][O.], C["GeometricQ uantity"][C["Unspecified"]["radius", "{{C[\"GeometricPoint\"][A.], C[\"G eometricPoint\"][C[\"Unspecifi ed\"][\"ref\", \"{2, 1, 1, 2, 0}\", \"1\"]], C[\"GeometricP oint\"][C[\"Unspecified\"][\"r ef\", \"{2, 1, 1, 2, 0}\", \"2\"]]}, {2, 1, 1, 2, 0}}"]]]}]{-7.46202, 1.41653}, GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["GeometricP oint"][C["Unspecified"]["ref", "{1, 1, 0}", "1"]], C["GeometricPoint" ][C["Unspecified"]["ref", "{1, 1, 0}", "2"]]}, {1, 1, 0}}] 7.83239, GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["GeometricP oint"][C["Unspecified"]["ref", "{2, 1, 1, 2, 0}", "1"]], C["Geometric Point"][C["Unspecified"]["ref", "{2, 1, 1, 2, 0}", "2"]]}, {2, 1, 1, 2, 0}}]7.83239, GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["GeometricP oint"][C["Unspecified"]["ref", "{3, 1, 2, 0}", "1"]], C["GeometricPoi nt"][C["Unspecified"]["ref", "{3, 1, 2, 0}", "2"]]}, {3, 1, 2, 0}}] 7.83239, GeometricQuantity Unspecified Angle[{C["GeometricPoint"][C.], C["GeometricPoint"][A.], C["GeometricPoint"][H.]}]],geoPlanarAngl e[{C["GeometricPoint"][D.], C["GeometricPoint"][E.], C["GeometricPoint"][F.]}]0.138369, GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["GeometricP oint"][C["Unspecified"]["ref", "{4, 1, 2, 0}", "1"]], C["GeometricPoi nt"][C["Unspecified"]["ref", "{4, 1, 2, 0}", "2"]]}, {4, 1, 2, 0}}] 7.83239, GeometricQuantity Unspecified Angle[{C["GeometricPoint"][B.], C["GeometricPoint"][A.], C["GeometricPoint"][G.]}]],geoPlanarAngl e[{C["GeometricPoint"][D.], C["GeometricPoint"][F.], C["GeometricPoint"][E.]}]0.852616 | |||||
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Additional instances
Dependency graphs
Shortest chains of proofs, from the axioms.
Direct dependencies.
Full dependencies, all the way down to the axioms.
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