Sum of Two Sides of Triangle Greater than Third Side

Theorem
Given a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side.

Proof

 * Euclid-I-20.png

Let $ABC$ be a triangle.

By, we can produce $BA$ past $A$ in a straight line.

By Construction of Equal Straight Lines from Unequal, there exists a point $D$ such that $DA = CA$.

Therefore, from Isosceles Triangle has Two Equal Angles:
 * $\angle ADC = \angle ACD$

Thus by :
 * $\angle BCD > \angle BDC$

Thus $\triangle DCB$ is a triangle having $\angle BCD$ greater than $\angle BDC$

Hence from Greater Angle of Triangle Subtended by Greater Side:
 * $BD > BC$

But:
 * $BD = BA + AD$

and:
 * $AD = AC$

Thus:
 * $BA + AC > BC$

A similar argument shows that $AC + BC > BA$ and $BA + BC > AC$.