Euclid's Elements- Book I- Proposition 20
If any triangle two sides taken together in any manner are greater than the remaining one.
In other words...
The sum of any two sides of a triangle is greater than the third side
1. Construct line AB.
2. Construct line AC.
3. Construct line BC.
Triangle ABC is a trilateral figure contained by three lines AB, BC, and AC by definition 19.
We want to show
i. AB+AC > BC
ii. AB+BC > AC
iii. BC+ AC > AB
4-5. Construct segment AD on line AB such that AD=AC.
6. Construct segment CD.
Triangle ACD is contained by lines AC, AD, and CD where AD=AC by construction.
By definition 20, triangle ACD is isosceles.