A history of Greek mathematics - Wilbourhall.org

366 PAPPUS OF ALEXANDRIA
triangle with base less than one of the other sides, it is possible
to construct on the base and within the triangle two straight
lines meeting at a point, the sum of which is equal to the sum
of the other two sides of the triangle (Props. 29, 30).
2. In any triangle in which it is possible to construct two
straight lines from the base to one internal point the sum
of which is equal to the sum of the two sides of the triangle,
it is also possible to construct two other such straight lines the
sum of which is greater than that sum (Prop. 31).
3. Under the same conditions, if the base is greater than either
of the other two sides, two straight lines can be so constructed
from the base to an internal point which are respectively
greater than the other two sides of the triangle ; and the lines
may be constructed so as to be respectively equal to the two
sides, if one of those two sides is less than the other and each
of them is less than the base (Props. 32, 33).
4. The lines may be so constructed that their sum will bear to
the sum of the two sides of the triangle any ratio less
2 : 1 (Prop. 34).
than
As examples of the proofs, we will take the case of the
scalene triangle, and prove the first and Part 1
the above propositions for such a triangle.
than
of the third of
In the triangle ABC with base BC let AB he greater
AC.
Take D on BA such that BD = J (BA + AC).
B H L
On DA between D and A take any point E, and draw EF
parallel to BC. Let G be any point on EF; draw GH parallel
to AB and join GC.