[kurthr]

Ok, I have read the first 20 slides for a CS class at an institution from which I have a degree. I feel no closer to understanding why it is relevant.

Can you specify which step in the "new proof" is wrong, unproven, or tautological (itself requires the Pythagorean theorem)?

The simplest reason why it is explanatory as defined in your cited reference is that the Euclidean distance is defined by the hypotenuse of the triangle, that triangle (and similar triangles by construction) have ratios which are also similar, the distance along a side is the sum of those distances in an infinite series, the series expansions of sin and cosine are known along with the law of sines and are independently (from the Pythagorean) proven.