Kitap 1
19
Neither is AC less than AB, for then the angle ABC would also have been less than the angle ACB; [I. 18] but it is not; therefore AC is not less than AB.
19
And it was proved that it is not equal either. Therefore AC is greater than AB.
19
Therefore etc.
19
Q. E. D.
Proposition 20.
20
Enunciation In any triangle two sides taken together in any manner are greater than the remaining one.
20
Proof. For let ABC be a triangle; I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely BA, AC greater than BC, AB, BC greater than AC, BC, CA greater than AB.
20
For let BA be drawn through to the point D, let DA be made equal to CA, and let DC be joined.
20
Then, since DA is equal to AC, the angle ADC is also equal to the angle ACD; [I. 5] therefore the angle BCD is greater than the angle ADC. [C.N. 5]
20
And, since DCB is a triangle having the angle BCD greater than the angle BDC, and the greater angle is subtended by the greater side, [I. 19] therefore DB is greater than BC.
20
But DA is equal to AC; therefore BA, AC are greater than BC.
20
Similarly we can prove that AB, BC are also greater than CA, and BC, CA than AB.
20
Therefore etc.
20
Q. E. D.
Proposition 21.
21
Enunciation If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.
21
Proof. On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be constructed meeting within the triangle;
21
I say that BD, DC are less than the remaining two sides of the triangle BA, AC, but contain an angle BDC greater than the angle BAC.
21
For let BD be drawn through to E.
21
Then, since in any triangle two sides are greater than the remaining one, [I. 20] therefore, in the triangle ABE, the two sides AB, AE are greater than BE.
21
Let EC be added to each; therefore BA, AC are greater than BE, EC.
