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 than 
2 : 1 (Prop. 34). 
As examples of the proofs, we will take the case of the 
scalene triangle, and prove the first and Part 1 of the third of 
the above propositions for such a triangle. 
In the triangle ABC with base BO let AB be greater 
than AC. 
Take D on BA such that BD = \ (BA +AC). 
A 
On DA between D and A take any point E, and draw EF 
parallel to BG. Let G be any point on EF] draw OH parallel 
to AB and join GC.