Euclid Book 6 Proposition 14a
Statement
Two equal area parallelograms (ABCD, EFGH) with a pair of equal angles (∠BAD∠FEH), have the sides adjacent to the equal angles inversely proportional ().
AB
EF
EH
AD
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.},{}},{Parallelogram[{A.,B.,C.,D.}],Parallelogram[{E.,F.,G.,H.}],PlanarAngle[{B.,A.,D.}]PlanarAngle[{F.,E.,H.}],Area[Polygon[{A.,B.,C.,D.}]]Area[Polygon[{E.,F.,G.,H.}]]},
EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[E.,H.]
EuclideanDistance[A.,D.]
EuclideanDistance[A.,B.] EuclideanDistance[E.,F.] EuclideanDistance[E.,H.] EuclideanDistance[A.,D.] |
 |
Explanations
Let , be equal and equiangular parallelograms having the angles at equal, and let , be placed in a straight line; therefore , are also in a straight line.
I say that, in, , the sides about the equal angles are reciprocally proportional, that is to say, that, as is to , so is to . For let the parallelogram be completed.
Since, then, the parallelogram is equal to the parallelogram , and is another area, therefore, as is to , so is to .
But, as is to , so is to ,and, as is to , so is to .[id.]therefore also, as is to , so is to .
Therefore in the parallelograms, the sides about the equal angles are reciprocally proportional.
AB
BC
B
DB
BE
FB
BG
I say that, in
AB
BC
DB
BE
GB
BF
FE
Since, then, the parallelogram
AB
BC
FE
AB
FE
BC
FE
But, as
AB
FE
DB
BE
BC
FE
GB
BF
DB
BE
GB
BF
Therefore in the parallelograms
AB
BC