Proof
Repeated application of I.38 generates any multiple of a triangle on
the corresponding multiple of the base; the resulting equimultiples
test (V.5) is exactly the statement of proportionality. The
parallelogram version follows because each parallelogram is double
its diagonal triangle (I.34, I.41).
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Depends on (4)
- I.34Proposition I.34In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
- I.38Proposition I.38Triangles which are on equal bases and in the same parallels are equal to one another.
- I.41Proposition I.41If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the…
- V.5Definition V.5Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any…
Required by (dependents) (6)
- VI.2Proposition VI.2If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle…
- VI.14Proposition VI.14In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular…
- VI.19Proposition VI.19Similar triangles are to one another in the duplicate ratio of the corresponding sides.
- VI.23Proposition VI.23Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.
- XI.25Proposition XI.25If a parallelepipedal solid be cut by a plane parallel to opposite planes, then, as the base is to the base, so will…
- XII.4Proposition XII.4If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids…
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