Book 6
DEFINITIONS.
1
1
Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.
2
2
[Reciprocally related figures. See note.]
3
3
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
4
4
The height of any figure is the perpendicular drawn from the vertex to the base.
5
BOOK VI. PROPOSITIONS.
PROPOSITION 1.
1
Triangles and parallelograms which are under the same height are to one another as their bases.
1
Let ABC, ACD be triangles and EC, CF parallelograms under the same height; I say that, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF.
1
For let BD be produced in both directions to the points H, L and let [any number of straight lines] BG, GH be made equal to the base BC, and any number of straight lines DK, KL equal to the base CD; let AG, AH, AK, AL be joined.
1
Then, since CB, BG, GH are equal to one another, the triangles ABC, AGB, AHG are also equal to one another. [I. 38]
1
Therefore, whatever multiple the base HC is of the base BC, that multiple also is the triangle AHC of the triangle ABC.
1
For the same reason, whatever multiple the base LC is of the base CD, that multiple also is the triangle ALC of the triangle ACD; and, if the base HC is equal to the base CL, the triangle AHC is also equal to the triangle ACL, [I. 38] if the base HC is in excess of the base CL, the triangle AHC is also in excess of the triangle ACL, and, if less, less.
