Real analysis – How to calculate $ pi $?

One of the most powerful algorithms to date to calculate. $ pi $
is the algorithm of Brent-Salamin (or Gauss-Legendre-Euler). Impressively
this algorithm that only uses a very simple arithmetic-geometric mean
iteration gets a quadratic convergence (the number
of the exact double digit in each iteration).

The average idea of ​​the algorithm are the two algebraic relationships:

With $$ I (a, b) = int_ {0} ^ { pi / 2} frac {1} { sqrt {a ^ {2} cos ^ {2} (t) + b ^ {2} sin ^ {2} (t)}} dt $$
Y $$ J (a, b) = int_ {0} ^ { pi / 2} sqrt {a ^ {2} cos ^ {2} (t) + b ^ {2} sin ^ {2} (t)} dt $$
Y $ a_ {n + 1} = frac {a_ {n} + b_ {n}} {2} $, $ b_ {n + 1} = sqrt {a_ {n} b_ {n}} $.

We have

one-$$ I (a_ {n + 1}, b_ {n + 1}) = I (a_ {n}, b_ {n}) $$

two-$$ J (a_ {n}, b_ {n}) = 2J (a_ {n + 1}, b_ {n + 1}) – a_ {n} b_ {n} I (a_ {n + 1}, b_ {n + 1}) $$
Then how $ a_ {n}, b_ {n} rightarrow l ^ {*} $ one can easily calculate
$ I (l ^ {*}, l ^ {*}) $ Y $ J (l ^ {*}, l ^ {*}) $.

I would like to understand how to prove these relationships. It seems
that we can do it with a little bit of algebraic manipulation that are not
Too hard but maybe a little tedious. I wonder if there are more
In a simple or intuitive way (geometric?).