G.241 segment apply parallelogram

VI.29

To a given segment to apply a parallelogram equal to a given polygon and exceeding by a parallelogrammic figure similar to a given one.

segment_apply_parallelogram_vi_29(/ A B /, < C >, < D >) → < A O >, < Q P >

given / A B / , < C > the given rectilineal figure to which the figure to be applied to / A B / is required to be equal, < D > that to which the excess is required to be similar

construct to / A B / there has been applied the parallelogram < A O > equal to the given rectilineal figure < C > and exceeding by a parallelogrammic figure < Q P > which is similar to < D >, since < P Q > is also similar to < E L >

given / A B / , < C > the given rectilineal figure to which the figure to be applied to / A B / is required to be equal, < D > that to which the excess is required to be similar

construct construct_figure_similar: Let / A B / be bisected at E ; let there be described on / E B / the parallelogram < B F > similar and similarly situated to < D > ; and let < G H > be constructed at once equal to the sum of < B F >, < C > and similar and similarly situated to < D >.

assert Let / K H / correspond to / F L / and / K G / to / F E /

assert < G H > > < F B >,

assert / K H / is also greater than / F L /, and / K G / than / F E /

construct Let [ F L ], [ F E ] be produced, let / F L M / be equal to / K H /, and / F E N / to / K G /, and let < M N > be completed ;

assert < M N > is both equal and similar to < G H >

assert < G H > is similar to < E L >

∵ figures_similar_polygonassert < M N > is also similar to < E L >

∵ parallelogram_there_takenassert < E L > is about the same diameter with < M N >

construct Let their diameter / F O / be drawn, and let the figure be described.

assert < G H > is equal to < E L >, < C >, while < G H > is equal to < M N >,

assert < M N > is also equal to < E L >, < C >

construct Let < E L > be subtracted from each ;

assert the remainder, the gnomon < X W V >, is equal to < C >

assert / A E / = / E B /,

∵ parallelograms_equal_bases, parallelogram_complementsassert < A N > is also equal to < N B >, that is, to < L P >

construct Let < E O > be added to each ;

assert < A O > is equal to the gnomon < V W X >

assert The gnomon < V W X > is equal to < C >

assert < A O > is also equal to < C >

∵ parallelogram_parallelograms_diameter∴ to / A B / there has been applied the parallelogram < A O > equal to the given rectilineal figure < C > and exceeding by a parallelogrammic figure < Q P > which is similar to < D >, since < P Q > is also similar to < E L >

Dependency Graph

Required for

G.243