Proving that the sum of any two sides of the triangle is greater than the third side

If they weren't, then the two sides would not manage to reach each other on paper. Draw a line and two circles with mid points at the end points of the line, if the sum of radiuses are not greater than the length of the line the circles will never intersect and you can not build a triangle.

Now behold my godly paint skills. If $c+b \lt a$ then no matter how we choose angles rotating around the end points of $a$ will we be able to make $b$ and $c$ ends meet to form a triangle.

A classic application of the incircle:

Use cosine law

$a^2=b^2 + c^2 -2bc \cos(A) \lt b^2 + c^2 + 2bc=(b+c)^2$ because $-1 \lt \cos(A) \lt 1$