Euclid Book 6 Proposition 16b
Statement
If the areas of two rectangles (ABCD, EFGH) are equal, then the corresponding sides of those rectangles are inversely proportional ().
AB
FG
EF
BC
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.},{a.,b.,c.,d.}},{GeometricAssertion[{Polygon[{A.,B.,C.,D.}],Polygon[{E.,F.,G.,H.}]},"Equiangular"],EuclideanDistance[A.,B.]a.,EuclideanDistance[B.,C.]b.,EuclideanDistance[E.,F.]c.,EuclideanDistance[F.,G.]d.,Area[Polygon[{A.,B.,C.,D.}]]Area[Polygon[{E.,F.,G.,H.}]],a.b.c.d.},,
EuclideanDistance[A.,B.]
EuclideanDistance[F.,G.]
EuclideanDistance[E.,F.]
EuclideanDistance[B.,C.]
a.
d.
c.
b.
EuclideanDistance[A.,B.] EuclideanDistance[F.,G.] EuclideanDistance[E.,F.] EuclideanDistance[B.,C.] |
 |
Explanations
Let the rectangle contained by , be equal to the rectangle contained by , ; I say that the four straight lines will be proportional, so that, as is to , so is to .
For, with the same construction, since the rectangle, is equal to the rectangle , , and the rectangle , is , for is equal to , and the rectangle , is , for is equal to , therefore is equal to .
And they are equiangular.
But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional.
Therefore, as is to , so is to .
But is equal to , and to ; therefore, as is to , so is to.
AB
F
CD
E
AB
CD
E
F
For, with the same construction, since the rectangle
AB
F
CD
E
AB
F
BG
AG
F
CD
E
DH
CH
E
BG
DH
And they are equiangular.
But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional.
Therefore, as
AB
CD
CH
AG
But
CH
E
AG
F
AB
CD
E
F