PROPOSITION 9. PROBLEM.
From a given straight line to cut off any part required.
Let AB be the given straight line: it is required to cut off any part from it.
From the point A draw a straight line AG, making any angle with AB;
in AG take any point D, and take AC the same multiple of AD, that AB is of the part which is to be cut off from it; join BC, and draw DE parallel to it. AE shall be the part required to be cut off.
For, because ED is parallel to BC, [Construction.
one of the sides of the triangle ABC,
therefore CD is to DA as BE is to EA; [VI. 2.
and, by composition, CA is to AD as BA is to AE. [V. 18.
But CA is a multiple of AD; [Construction.
therefore BA is the same multiple of AE; [V. D.
that is, whatever part AD is of AC, AE is the same part of AB.
Wherefore, from the given straight line AB, the part required has been cut off. q.e.f.
PROPOSITION 10. PROBLEM.
To divide a given straight line similarly to a given divided straight line, that is, into parts which shall have the same ratios to one another, that the parts of the given divided straight line have.
Let AB be the straight line given to be divided, and AC the given divided straight line: it is required to divide AB similarly to AC.
Let AC be divided at the points D, E and let AB, AC be placed so as to contain any angle, and join BC; through the point D, draw DF parallel to BC, and through the point E draw EG parallel to BC. [I. 31.
AB shall be divided at the points F, G, similarly to AC.
