vector addition
Below you'll see a vector x and a vector y. Verify for yourself that the vector indicated by x+y is indeed properly labeled.
Move the points B and C around; that will change the vectors x and y. What do you notice about the relative lengths of the vectors?
Now, take a look at Lemma 1.8.
Make sure that you believe the labels in the sketch below.
Now, see what Lemma 1.10 says about the lengths of these vectors. Can you see why it's called the "Parallelogram Identity"?