Proof
Reduce both given figures to rectangles on a common base (I.44);
the side opposite the common base measures each figure's area. Take
the mean proportional (VI.13) of those two opposite sides; build on
that mean a figure similar to the first via VI.18. By VI.20 the
constructed figure has the required area.
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Depends on (4)
- I.44Proposition I.44To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.
- VI.13Proposition VI.13To two given straight lines to find a mean proportional.
- VI.18Proposition VI.18On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.
- VI.20Proposition VI.20Similar polygons are divided into similar triangles, equal in multitude and in the same ratio as the wholes; and the…
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