G.581 segment cut extreme

XIII.3

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

segment_cut_extreme_xiii_3()

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

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

assert Let / A C / be the greater segment, and let / A C / be bisected at D

construct Let the square < A E > be described on / A B /

construct Let the figure be drawn double.

assert / A C / is double of / D C /,

assert the square on / A C / is quadruple of the square on / D C /, that is, < R S > is quadruple of < F G >

assert since the rectangle / A B /, / B C / is equal to the square on / A C /, and < C E > is the rectangle / A B /, / B C /,

assert < C E > is equal to < R S >

assert < R S > is quadruple of < F G >

assert < C E > is also quadruple of < F G >

assert / A D / = / D C /,

assert / H K / = / K F /

assert the square < G F > is also equal to the square < H L >

assert / G K / = / K L /, that is, / M N / to / N E / ; hence < M F > is also equal to < F E >

assert < M F > is equal to < C G >

assert < C G > is also equal to < F E >

construct Let < C N > be added to each ;

assert the gnomon < O P Q > is equal to < C E >

assert < C E > was proved quadruple of < G F >

assert the gnomon < O P Q > is also quadruple of the square < F G >

assert the gnomon < O P Q > and the square < F G > are five times < F G >

assert The gnomon < O P Q > and the square < F G > are the square < D N >

assert < D N > is the square on / D B /, and < G F > the square on / D C /

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

Dependency Graph

Required for

G.597, G.601