Euclid Book 6 Proposition 16a
Statement
If the corresponding sides of two rectangles (ABCD, EFGH) are inversely proportional (), then the areas of those rectangles are equal.
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.}
a.
d.
c.
b.
Area[Polygon[{A.,B.,C.,D.}]]Area[Polygon[{E.,F.,G.,H.}]] |
 |
Explanations
Let the four straight lines , , , be proportional, so that, as is to , so is to ; I say that the rectangle contained by , is equal to the rectangle contained by , .
Let, be drawn from the points , at right angles to the straight lines , , and let be made equal to , and equal to . Let the parallelograms , be completed.
Then since, as is to , so is to, while is equal to , and to , therefore, as is to , so is to .
Therefore in the parallelograms, the sides about the equal angles are reciprocally proportional.
But those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal; therefore the parallelogram is equal to the parallelogram .
And is the rectangle , , for is equal to ; and is the rectangle , , for is equal to ; therefore the rectangle contained by , is equal to the rectangle contained by , .
AB
CD
E
F
AB
CD
E
F
AB
F
CD
E
Let
AG
CH
A
C
AB
CD
AG
F
CH
E
BG
DH
Then since, as
AB
CD
E
F
E
CH
F
AG
AB
CD
CH
AG
Therefore in the parallelograms
BG
DH
But those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal; therefore the parallelogram
BG
DH
And
BG
AB
F
AG
F
DH
CD
E
E
CH
AB
F
CD
E