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 1,2,1,11,1,1,2,2,13,1,2,2,1,1...
What comes next?