Euclid Book 6 Proposition 24
Statement
If two parallelograms (ABCD, EFGH) have parallel corresponding sides and their diagonals (, ) lie on the same line, then those parallelograms are similar.
AC
EG
Computational Explanation
GeometricScene[{{A.,B.,C.,D.,E.,F.,G.,H.},{}},{Parallelogram[{A.,B.,C.,D.}],Parallelogram[{E.,F.,G.,H.}],GeometricAssertion[{{Line[{A.,B.}],Line[{E.,F.}]},{Line[{B.,C.}],Line[{F.,G.}]}},"Parallel"],Line[{{A.,C.},{E.,G.}}],GeometricAssertion[{A.,C.,E.,G.},"Collinear"]},{GeometricAssertion[{Parallelogram[{A.,B.,C.,D.}],Parallelogram[{E.,F.,G.,H.}]},"Similar"]}]
GeometricAssertion[{Parallelogram[{A.,B.,C.,D.}],Parallelogram[{E.,F.,G.,H.}]},Similar] |
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Explanations
Let be a parallelogram, and its diameter, and let , be parallelograms about ; I say that each of the parallelograms , is similar both to the whole and to the other.
For, since has been drawn parallel to , one of the sides of the triangle , proportionally, as is to , so is to .
Again, since has been drawn parallel to , one of the sides of the triangle , proportionally, as is to , so is to .
But it was proved that, as is to , so also is to ; therefore also, as is to , so is to , and therefore, componendo, as is to , so is to ,and, alternately, as is to , so is to .
Therefore in the parallelograms, , the sides about the common angle are proportional.
And, since is parallel to , the angle is equal to the angle ; and the angle is common to the two triangles , ; therefore the triangle is equiangular with the triangle .
For the same reason the triangle is also equiangular with the triangle , and the whole parallelogram is equiangular with the parallelogram .
Therefore, proportionally, as is to , so is to , as is to , so is to , as is to , so is to , and further, as is to , so is to .
And, since it was proved that, as is to , so is to , and, as is to , so is to , therefore, ex aequali, as is to , so is to .
Therefore in the parallelograms, the sides about the equal angles are proportional; therefore the parallelogram is similar to the parallelogram .
For the same reason the parallelogram is also similar to the parallelogram ; therefore each of the parallelograms , is similar to .
But figures similar to the same rectilineal figure are also similar to one another; therefore the parallelogram is also similar to the parallelogram .
ABCD
AC
EG
HK
AC
EG
HK
ABCD
For, since
EF
BC
ABC
BE
EA
CF
FA
Again, since
FG
CD
ACD
CF
FA
DG
GA
But it was proved that, as
CF
FA
BE
EA
BE
EA
DG
GA
BA
AE
DA
AG
BA
AD
EA
AG
Therefore in the parallelograms
ABCD
EG
BAD
And, since
GF
DC
AFG
DCA
DAC
ADC
AGF
ADC
AGF
For the same reason the triangle
ACB
AFE
ABCD
EG
Therefore, proportionally, as
AD
DC
AG
GF
DC
CA
GF
FA
AC
CB
AF
FE
CB
BA
FE
EA
And, since it was proved that, as
DC
CA
GF
FA
AC
CB
AF
FE
DC
CB
GF
FE
Therefore in the parallelograms
ABCD
EG
ABCD
EG
For the same reason the parallelogram
ABCD
KH
EG
HK
ABCD
But figures similar to the same rectilineal figure are also similar to one another; therefore the parallelogram
EG
HK