G.490 segment commensurable apotome

X.104

A segment commensurable with an apotome of a medial segment is an apotome of a medial segment and the same in order.

segment_commensurable_apotome(/ A B /, / C D /)

given / A B / be an apotome of a medial segment, let / C D / be commensurable in length with / A B /

prove / C D / is also an apotome of a medial segment and the same in order with / A B /

given / A B / be an apotome of a medial segment, let / C D / be commensurable in length with / A B /

assert / A B / is an apotome of a medial segment, let / E B / be the annex to it

∵ medial_segment_there, medial_segment_there_x_75assert / A E /, / E B / are medial segments commensurable in square only

∵ fourth_proportionalassert Let it be contrived that, as / A B / is to / C D /, so is / B E / to / D F /

∵ number_magnitudes_proportional, magnitudes_proportional_commensurableassert / A E / is also commensurable with / C F /, and / B E / with / D F /

assert / A E /, / E B / are medial segments commensurable in square only

∵ segment_commensurable_medial, magnitudes_commensurable_incommensurable, medial_segment_there, medial_segment_there_x_75assert / C F /, / F D / are also medial segments commensurable in square only; therefore / C D / is an apotome of a medial segment

assert I say next that it is also the same in order with / A B /

assert Since, as / A E / is to / E B /, so is / C F / to / F D /,

assert also, as the square on / A E / is to the rectangle / A E /, / E B /, so is the square on / C F / to the rectangle / C F /, / F D /

assert The square on / A E / is commensurable with the square on / C F /

∵ magnitudes_proportional_proportional, magnitudes_proportional_commensurableassert the rectangle / A E /, / E B / is also commensurable with the rectangle / C F /, / F D /

∵ square_assigned_segment_def, segment_commensurable_medialassert if the rectangle / A E /, / E B / is rational, the rectangle / C F /, / F D / will also be rational, and if the rectangle / A E /, / E B / is medial, the rectangle / C F /, / F D / is also medial

∵ medial_segment_there, medial_segment_there_x_75∴ / C D / is an apotome of a medial segment and the same in order with / A B /

Dependency Graph