Commentary on Proposition VI.12¶
Back to Proposition VI.12.
We have here the geometrical equivalent of the “rule of three.”
It is of course immaterial whether, as in Euclid’s proof, the first and second straight lines are measured on one of the lines forming the angle and the third on the other, or the first and third are measured on one and the second on the other.
If it should be desired that the first and the required fourth be measured on one of the lines, and the second and third on the other, we can use the following construction. Measure DE on one straight line equal to A, and on any other straight line making an angle with the first at the point D measure DF equal to B, and DG equal to C. Join EF, and through G draw GH anti-parallel to EF, i.e. make the angle DGH equal to the angle DEF; let GH meet DE (produced if necessary) in H.
DH is then the fourth proportional.
For the triangles EDF, GDH are similar, and the sides about the equal angles are proportional, so that
or