Euclid Book 6 Proposition 17b
Statement
If a rectangle (ABCD) and a square (EFGH) have equal areas, then one side () of the rectangle is to the side () of the square as the side of the square is to the other side () of the rectangle ().
AB
EF
BC
AB
EF
EF
BC
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.},{a.,b.,c.}},GeometricAssertion[{Polygon[{A.,B.,C.,D.}]},"Equiangular"],GeometricAssertion[{Polygon[{E.,F.,G.,H.}]},"Regular"],EuclideanDistance[A.,B.]a.,EuclideanDistance[B.,C.]b.,EuclideanDistance[E.,F.]c.,Area[Polygon[{A.,B.,C.,D.}]]Area[Polygon[{E.,F.,G.,H.}]],a.b.,,
2
c.
EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[E.,F.]
EuclideanDistance[B.,C.]
a.
c.
c.
b.
EuclideanDistance[A.,B.] EuclideanDistance[E.,F.] EuclideanDistance[E.,F.] EuclideanDistance[B.,C.] |
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Explanations
Let the rectangle , be equal to the square on ; I say that, as is to , so is to .
For, with the same construction, since the rectangle, is equal to the square on , while the square on is the rectangle , , for is equal to , therefore the rectangle , is equal to the rectangle , .
But, if the rectangle contained by the extremes be equal to that contained by the means, the four straight lines are proportional.
Therefore, as is to , so is to .
But is equal to ; therefore, as is to , so is to .
A
C
B
A
B
B
C
For, with the same construction, since the rectangle
A
C
B
B
B
D
B
D
A
C
B
D
But, if the rectangle contained by the extremes be equal to that contained by the means, the four straight lines are proportional.
Therefore, as
A
B
D
C
But
B
D
A
B
B
C