Hello,

to #1:

1. Draw a sketch.

2. Inflect the triangle ABC over the point D (not sure if this is the correct expression; see attachment)

3. You'll get a parallelogram ABA'C. The diagonal of the parallelogram is 2*(AD)

4. Consider the triangle ABA' with BA' = AC

5. According to the triangle inequality

AB + BA' > AA' = 2*(AD)

AB + AC > 2*(AD)