Proof
By VI.8 the altitude from the right angle cuts the hypotenuse into
segments such that each leg is the mean proportional between the
hypotenuse and the adjacent segment. Applying VI.20 (areas of
similar figures are in the duplicate ratio of corresponding sides)
gives each leg-figure equal to its adjacent piece of the
hypotenuse-figure. Summing the two pieces by V.24 yields the
hypotenuse-figure.
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Depends on (4)
- VI.8Proposition VI.8If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the…
- VI.19Proposition VI.19Similar triangles are to one another in the duplicate ratio of the corresponding sides.
- VI.20Proposition VI.20Similar polygons are divided into similar triangles, equal in multitude and in the same ratio as the wholes; and the…
- V.24Proposition V.24If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second…
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