Visualization for Euclidean Algorithm I want to really understand the Euclidean Algorithm.
A key component in the algorithm is fact that common divisors of two integers are common divisors of their difference.
I can see from the perspective of symbolic manipulation that for all common divisors of $a$ and $b$:
$d|a \land d|b$ means $a = kd \land b = ld \therefore a - b = kd - ld = (k - l)d$.
The trouble is, this doesn't convince me. I need am image or some way of grasping this beyond just the symbols.
The same for the rest of the algorithm. I get that the division/modulo version is really just applying subtraction multiple times, so understanding the subtraction part should help with that.
Then there is the issue of the last non-zero remainder being the GCD. Again, I can kind of understand via algebraic proof, but it's not convincing for me. I wouldn't bet much money on being able to convince an expert that I really understand.
So, any and all visual proofs, intuitions etc. that don't rely on that vertical list of divisors becoming dividends and remainders become divisors would be much appreciated.

For myself I found useful to visualize the steps and the various parameters that comes into
play by representing the Extended Euclidean Algorithm with the corresponding expansion into
continued fraction, for instance
$$
\eqalign{
  & {{34} \over {13}} = \left\lfloor {{{34} \over {13}}} \right\rfloor  + {8 \over {13}} = 2 + {1 \over {{{13} \over 8}}} = 2 + {1 \over {1 + {5 \over 8}}} = 2 + {1 \over {1 + {1 \over {{8 \over 5}}}}} =   \cr 
  &  = 2 + {1 \over {1 + {1 \over {1 + {3 \over 5}}}}} = 2 + {1 \over {1 + {1 \over {1 + {1 \over {{5 \over 3}}}}}}} = 2 + {1 \over {1 + {1 \over {1 + {1 \over {1 + {1 \over {{3 \over 2}}}}}}}}} =   \cr 
  &  = 2 + {1 \over {1 + {1 \over {1 + {1 \over {1 + {1 \over {1 + {1 \over 2}}}}}}}}} = 2 + {1 \over {1 + {1 \over {1 + {1 \over {1 + {1 \over {1 + {1 \over {1 + 1}}}}}}}}}} \cr} 
$$
