To cut a given finite straight line in extreme and mean ratio
be the given finite straight line; thus it is required to cut AB in extreme and mean ratio.
let the square BC
be described; and let there be applied to AC
the parallelogram CD
equal to BC
and exceeding by the figure AD
similar to BC
. [VI. 29
is a square; therefore AD is also a square.
And, since BC
is equal to CD
, let CE
be subtracted from each; therefore the remainder BF is equal to the remainder AD.
But it is also equiangular with it; therefore in BF
the sides about the equal angles are reciprocally proportional; [VI. 14
] therefore, as FE is to ED, so is AE to EB.
is equal to AB
, and ED
Therefore, as BA
is to AE
, so is AE
is greater than AE
; therefore AE is also greater than EB.
Therefore the straight line AB
has been cut in extreme and mean ratio at E
, and the greater segment of it is AE
. Q. E. F.