Euclid Book 6 Proposition 29
Statement
Given a parallelogram ECBD with C the midpoint of and a convex polygon GHIJ, to construct a parallelogram AZKX with the same area as GHIJ and which is the union of the parallelograms AZYB and BYKX, where BYKX is similar to ECBD.
AB
Computational Explanation
GeometricSceneA.,B.,C.,D.,E.,G.,H.,I.,J.,K.,M.,N.,P.,Q.,R.,S.,X.,Y.,Z.,{},{Style[Line[{A.,B.}],Blue],C.Midpoint[Line[{A.,B.}]],GeometricAssertion[{Parallelogram[{E.,C.,B.,D.}]},"Counterclockwise"],GeometricAssertion[{Style[Polygon[{G.,H.,I.,J.}],Yellow]},"Convex"]},GeometricAssertionParallelogram[P.,Q.,R.,S.],"Counterclockwise",GeometricAssertionPolygon[P.,Q.,R.,S.],Polygon[{E.,C.,B.,D.}],"Similar",AreaPolygon[P.,Q.,R.,S.]Area[Polygon[{E.,C.,B.,D.}]]+Area[Polygon[{G.,H.,I.,J.}]],Line[{E.,D.,M.}],EuclideanDistance[E.,M.]EuclideanDistance[P.,S.],Line[{E.,C.,N.}],EuclideanDistance[E.,N.]EuclideanDistance[P.,Q.],Parallelogram[{E.,N.,K.,M.}],{Line[{C.,B.,X.}],X.∈Line[{M.,K.}],Line[{D.,B.,Y.}],Y.∈Line[{N.,K.}]},{GeometricAssertion[{Line[{A.,Z.}],Line[{C.,N.}]},"Parallel"],GeometricAssertion[{Line[{A.,Z.}],InfiniteLine[{K.,N.}]},{"Concurrent",Z.}]},{Style[Polygon[{A.,Z.,K.,X.}],Pink],Style[Polygon[{B.,Y.,K.,X.}],Blue]},{Area[Polygon[{A.,Z.,K.,X.}]]Area[Polygon[{G.,H.,I.,J.}]],GeometricAssertion[{Polygon[{B.,Y.,K.,X.}],Polygon[{E.,C.,B.,D.}]},"Similar"]}
RandomInstanceGeometricSceneA.,B.,C.,D.,E.,G.,H.,I.,J.,K.,M.,N.,P.,Q.,R.,S.,X.,Y.,Z.,{},{Style[Line[{A.,B.}],Blue],C.Midpoint[Line[{A.,B.}]],GeometricAssertion[{Parallelogram[{E.,C.,B.,D.}]},Counterclockwise],GeometricAssertion[{Style[Polygon[{G.,H.,I.,J.}],Yellow]},Convex]},GeometricAssertionParallelogramP.,Q.,R.,S.,Counterclockwise,GeometricAssertionPolygonP.,Q.,R.,S.,Polygon[{E.,C.,B.,D.}],Similar,AreaPolygonP.,Q.,R.,S.Area[Polygon[{E.,C.,B.,D.}]]+Area[Polygon[{G.,H.,I.,J.}]],Line[{E.,D.,M.}],EuclideanDistance[E.,M.]EuclideanDistance[P.,S.],Line[{E.,C.,N.}],EuclideanDistance[E.,N.]EuclideanDistanceP.,Q.,Parallelogram[{E.,N.,K.,M.}],{Line[{C.,B.,X.}],X.∈Line[{M.,K.}],Line[{D.,B.,Y.}],Y.∈Line[{N.,K.}]},{GeometricAssertion[{Line[{A.,Z.}],Line[{C.,N.}]},Parallel],GeometricAssertion[{Line[{A.,Z.}],InfiniteLine[{K.,N.}]},{Concurrent,Z.}]},{Style[Polygon[{A.,Z.,K.,X.}],Pink],Style[Polygon[{B.,Y.,K.,X.}],Blue]},{Area[Polygon[{A.,Z.,K.,X.}]]Area[Polygon[{G.,H.,I.,J.}]],GeometricAssertion[{Polygon[{B.,Y.,K.,X.}],Polygon[{E.,C.,B.,D.}]},Similar]}
Explanations
coming soon