Euclid Book 3
Proposition 33
Construction
Previous:
Euclid Book 3 Proposition 32Next:
Euclid Book 3 Proposition 34Statement
On a given line segment to describe a segment of a circle which contains an angle equal to a given angle.
Construction steps
1. Let be the given line segment and ∠PDQ the given angle.
AB
2. Take A as a vertex, construct ∠BAE∠PDQ.
3. Let be perpendicular to at A, and perpendicular to at B.
CA
AE
CB
AB
4. Bisect at O.
AC
5. Construct the circle through the points AandC centered at O. On the given line segment , ∠ACB has been constructed equal to the given angle ∠PDQ.
AB
Original statement
ἐπὶ τῆς δοθϵίσης ϵὐθϵίας γράψαι τμῆμα κύκλου δϵχόμϵνον γωνίαν ἴσην τῇ δοθϵίσῃ γωνίᾳ ϵὐθυγράμμῳ.
English translation
On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle.
Computable version
GeometricSceneA.,B.,D.,P.,Q.,E.,C.,O.,{},GeometricStep{Line[{A.,B.}]},Line[{D.,P.},D.,Q.],PlanarAngle[P.,D.,Q.]<90°,GeometricStepLine[{A.,E.}],(*EuclideanDistance[A.,E.]<5,*)PlanarAngle[{B.,A.,E.}]PlanarAngle[P.,D.,Q.],GeometricStep[{GeometricAssertion[{Line[{C.,A.}],Line[{A.,E.}]},"Perpendicular"],GeometricAssertion[{Line[{C.,B.}],Line[{A.,B.}]},"Perpendicular"]}],GeometricStep[{O.Midpoint[Line[{A.,C.}]]}],GeometricStep[{CircleThrough[{A.,C.},O.]}],PlanarAngle[{A.,C.,B.}]PlanarAngle[P.,D.,Q.]//RandomInstance
Out[]=
| |||||
PlaneGeometry`GeometrySceneDrawerDump`makeGraph icsFromRulesAbsoluteThickness[2], EdgeForm try`GeometrySceneDrawerDump`makeAngleGraph ics{PlaneGeometry`GeometrySceneDrawerDump `geoAngle[ GeometricPoint GeometricPoint GeometricPoint Polygon[{ GeometricPoint GeometricPoint GeometricPoint ["GeometricPoint"][P.], C["GeometricPoint"][D.], C["GeometricPoint"][Q.]}],PlaneGeometry` GeometrySceneDrawerDump`makeAngleGraphicsPlaneGeometry`GeometrySceneDrawerDump`geoAngle GeometricPoint GeometricPoint GeometricPoint Polygon GeometricPoint GeometricPoint GeometricPoint e[{C["GeometricPoint"][P.], C["GeometricPoint"][D.], C["GeometricPoint"][Q.]}],PlaneGeometry` GeometrySceneDrawerDump`makeAngleGraphicsPlaneGeometry`GeometrySceneDrawerDump`geoAngle[ GeometricPoint GeometricPoint GeometricPoint Polygon[{ GeometricPoint GeometricPoint GeometricPoint aneGeometry`GeometrySceneDrawerDump`makeAn gleGraphicsPlaneGeometry`GeometrySceneDr awerDump`geoAngle GeometricPoint GeometricPoint GeometricPoint Polygon GeometricPoint GeometricPoint GeometricPoint 0.781163,{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 GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricPoint GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["GeometricPoint"][C.], C["Geometr icPoint"][C["Unspecified"]["ref", "{5, 1, 0}", "1"]]}, {5, 1, 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 GeometricPoint 4.73331}, GeometricPoint -3.43894}, GeometricPoint 1.0598}, GeometricPoint -4.09262}, GeometricPoint 1.5284}, GeometricPoint 2.89656}, GeometricPoint 2.23172}, GeometricPoint -0.279466}, GeometricQuantity Unspecified C["GeometricPoint"][D.], C["GeometricPoint"][Q.]}],geoPlanarAngl e[{C["GeometricPoint"][P.], C["GeometricPoint"][D.], C["GeometricPoint"][Q.]}]0.781163, GeometricQuantity Unspecified "GeometricPoint"][B.], C["GeometricPoint"][A.], C["GeometricPoint"][E.]}]],geoPlanarAngl e[{C["GeometricPoint"][P.], C["GeometricPoint"][D.], C["GeometricPoint"][Q.]}]0.781163, GeometricQuantity Unspecified {{C["GeometricPoint"][A.], C["GeometricPoint"][C.], C["GeometricP oint"][C["Unspecified"]["ref", "{5, 1, 0}", "1"]]}, {5, 1, 0}}] 6.61153 | |||||
|
Additional instances
Dependency graphs
Shortest chains of proofs, from the axioms.
Direct dependencies.
Full dependencies, all the way down to the axioms.
 |
      |