This is not really a puzzle, but a real geometry problem. Let's take a random triangle (ABC), and let's assume that the angle bisector from A intersects BC in the point D. Proof that:

   AD ^ 2 = AB * AC - BD * CD

Here is the figure, drawn in MSPAINT.EXE as you can see :-)

And now, a real math puzzle. Here is a sequence of sequences of numbers.

     1
    1,1
    2,1
  1,2,1,1
1,1,1,2,2,1
3,1,2,2,1,1
...

What comes next?