*BOOK I*. 19, 20, 21.

therefore *AC* is not less than *AB*.

And it has been shewn that *AC* is not equal to *AB*.

Therefore *AC* is greater than *AB*.

Wherefore, the *greater angle* &c. q.e.d.

PROPOSITION 20. *THEOREM*.

Any two sides of a triangle are together greater than the third side.

Let *ABC* be a triangle: any two sides of it are together greater than the third side; namely, *BA*, *AC* greater than *BC*; and *AB*, *BC* greater than *AC*; and *BC*, *CA* greater than *AB*.

Produce *BA* to *D*,

making *AD* equal to *AC*, [I. 3.

and join *DC*.

Then, because *AD* is equal to *AC*, [*Construction*.

the angle *ADC* is equal to the angle *ACD*. [I. 5.

But the angle *BCD* is greater than the angle *ACD*. [*Ax*. 9.

Therefore the angle *BCD* is greater than the angle *BDC*.

And because the angle *BCD* of the triangle *BCD* is greater than its angle *BDC*, and that the greater angle is subtended by the greater side; [I. 19.

therefore the side *BD* is greater than the side *BC*.

But *BD* is equal to *BA* and *AC*.

Therefore *BA*, *AC* are greater than *BC*.

In the same manner it may be shewn that *AB*, *BC* are greater than *AC*, and *BC*, *CA* greater than *AB*.

Wherefore, *any two sides* &c. q.e.d.

PROPOSITION 21. *THEOREM.*

If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.