and extending it once in the AB direction and twice in the AC direction,

We can start at the corner opposite to A of the parallelogram of which AB abd AC are adjacent sides (A+AB+AC)

From there, go to A+AB+AC+AB to A+AB+AC+AB+2BC to A+AB+AC+AB+2BC+3CA

and you will find yourself back at A, the other end of the diagonal.

The length of the diagonal is b²+c²+2bc.cos(A)