PROPOSITION 24.
In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.
Let
ABCD be a parallelogram, and
AC its diameter, and let
EG,
HK be parallelograms about
AC; I say that each of the parallelograms
EG,
HK is similar both to the whole
ABCD and to the other.
For, since
EF has been drawn parallel to
BC, one of the sides of the triangle
ABC,
proportionally, as BE is to EA, so is CF to FA. [VI. 2]
Again, since
FG has been drawn parallel to
CD, one of the sides of the triangle
ACD,
proportionally, as CF is to FA, so is DG to GA. [VI. 2]
But it was proved that,
as CF is to FA, so also is BE to EA; therefore also, as BE is to EA, so is DG to GA, and therefore,
componendo,
as BA is to AE, so is DA to AG, [V. 18] and, alternately,
as BA is to AD, so is EA to AG. [V. 16]
Therefore in the parallelograms
ABCD,
EG, the sides about the common angle
BAD are proportional.
And, since
GF is parallel to
DC,
the angle AFG is equal to the angle DCA; and the angle
DAC is common to the two triangles
ADC,
AGF;
therefore the triangle ADC is equiangular with the triangle AGF.
For the same reason
the triangle ACB is also equiangular with the triangle AFE, and the whole parallelogram
ABCD is equiangular with the parallelogram
EG.
Therefore, proportionally,
as AD is to DC, so is AG to GF, as DC is to CA, so is GF to FA, as AC is to CB, so is AF to FE, and further, as
CB is to
BA, so is
FE to
EA.
And, since it was proved that,
as DC is to CA, so is GF to FA, and, as
AC is to
CB, so is
AF to
FE, therefore,
ex aequali, as
DC is to
CB, so is
GF to
FE. [
V. 22]
Therefore in the parallelograms
ABCD,
EG the sides about the equal angles are proportional; therefore the parallelogram
ABCD is similar to the parallelogram
EG. [
VI. Def. 1]
For the same reason the parallelogram
ABCD is also similar to the parallelogram
KH; therefore each of the parallelograms
EG,
HK is similar to
ABCD.
But figures similar to the same rectilineal figure are also similar to one another; [
VI. 21] therefore the parallelogram
EG is also similar to the parallelogram
HK.
Therefore etc. Q. E. D.
