G.579 segment cut extreme

XIII.1

If a segment be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole is five times the square on the half.

segment_cut_extreme()

prove the square on / C D / is five times the square on / A D /

construct Let / A B / be cut in extreme and mean ratio at C

assert Let / A C / be the greater segment

construct extend(/ C A /) → / A D /

construct Let / A D / be made half of / A B /

construct Let the squares < A E >, < D F > be described on / A B /, / D C /, and let the figure in < D F > be drawn

assert Let / F C / be carried through to G

assert / A B / has been cut in extreme and mean ratio at C,

∵ segment_said_cut_def, segments_proportional_rectangle_vi_17assert the rectangle / A B /, / B C / is equal to the square on / A C /

assert < C E > is the rectangle / A B /, / B C /, and < F H > the square on / A C /

assert < C E > is equal to < F H >

assert since / B A / is double of / A D /, while / B A / = / K A /, and / A D / to / A H /,

assert / K A / is also double of / A H /

∵ triangles_parallelograms_heightassert But, as / K A / is to / A H /, so is < C K > to < C H >

assert < C K > is double of < C H >

assert < L H >, < H C > are also double of < C H >

assert < K C > is equal to < L H >, < H C >

assert < C E > was also proved equal to < H F >

assert the whole square < A E > is equal to the gnomon < M N O >

assert since / B A / is double of / A D /,

assert The square on / B A / is quadruple of the square on / A D /, that is, < A E > is quadruple of < D H >

assert < A E > is equal to the gnomon < M N O >

assert the gnomon < M N O > is also quadruple of < A P >; therefore < D F > is five times < A P >

assert < D F > is the square on / D C /, and < A P > the square on / D A /

∴ the square on / C D / is five times the square on / D A /

Dependency Graph

Required for

G.584, G.589, G.597, G.600, G.601