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?