Geometry problem solved three different ways
(www.youtube.com)https://www.youtube.com/watch?v=AZt70Ob6bFkSecondary school mathematics emphasizes method. Here is
the way to do it. This spills over into proof. We ask "what is
the proof". I like this video because the problem is not too hard, there seems to be little room for alternative approaches. Yet the video solves the problem in three
distinctly different ways. That pushes back against the error of thinking that there is only one method and only one proof.
I also liked this video because the problem comes up later in trying to compute pi. One idea for computing pi is that tan(pi/4) = 1. That suggests a plan: find a power series for arctan and use it to evaluate pi = 4 arctan(1).
Since arctan is the integral of 1(1 + x^2) we can use the power series 1/(1 + x^2) = 1 - x^2 + x^4 - x^6 + x^8 - ...
Integrate term by term to get a power series for arctan. Evaluate at 1. Notice that 1 is the radius of convergence of the power series. It converges at 1, but slowly. The plan hasn't really worked.
But, spoiler ahead, what if we had some clever identity such as pi/4 = arctan(1/2) + arctan(1/3)? We would have to evaluate the power series twice, for 1/2 and for 1/3, but both converge geometrically; that makes a much better plan, and many more significant digits for our labor. The video doesn't go there, but the problem it solves is the problem of the clever identity that you will want later for computing pi
