*EUCLID'S ELEMENTS.*

Let *ABC* be a triangle, and from the points *B*, *C*, the ends of the side *BC*, let the two straight lines *BD*, *CD* be drawn to the point *D* within the triangle: *BD*, *DC* shall be less than the other two sides *BA*, *AC* of the triangle, but shall contain an angle *BDC* greater than the angle *BAC*.

Produce *BD* to meet *AC* at *E*.

Because two sides of a triangle are greater than the third side, the two sides *BA*, *AE* of the triangle *ABE* are greater than the side *BE*. [I. 20.

To each of these add *EC*.

Therefore *BA*, *AC* are greater than *BE*, *EC*.

Again; the two sides *CE*, *ED* of the triangle *CED* are greater than the third side *CD*. [I. 20.

To each of these add *DB*.

Therefore *CE*, *EB* are greater than *CD*, *DB*.

But it has been shewn that *BA*, *AC* are greater than *BE*, *EC*;

much more then are *BA*, *AC* greater than *BD*, *DC*.

Again, because the exterior angle of any triangle is greater than the interior opposite angle, the exterior angle *BDC* of the triangle *CDE* is greater than the angle *CED*. [I. 16.

For the same reason, the exterior angle *CEB* of the triangle *ABE* is greater than the angle *BAE*.

But it has been shewn that the angle *BDC* is greater than the angle *CEB*;

much more then is the angle *BDC* greater than the angle *BAC*.

Wherefore, *if from the ends* &c. q.e.d.