I enjoy this nice site very much.

After reading about Dijkstra and the Pythagorean Theorem
( https://www.cut-the-knot.org/pythagoras/Dijkstra.shtml )
I realised that the problem with his formula
sgn(\alpha+\beta-\gamma)=sgn(a^2+b^2-c^2)
is that it involves negative quantities. One should not construct the difference of angles, but their sum. For the picture this means placing the two smaller triangles on the outside. In the right-angled case one gets the figure of Proof #41. Computing the side lengths as in that Proof, one gets a simple proof of Dijkstra's formula.
The formula is a qualitative version of the more precise quantitative formula
a^2+b^2-c^2= 2ab\sin 1/2(\alpha+\beta-\gamma)
which is basically the cosine rule. One can use the same picture to prove the rule.

I have written a small text, with pictures, detailing the above. It is on my home page
https://www.math.chalmers.se/~stevens/dijkstra.pdf

Best wishes,
Jan Stevens
Matematik, Göteborgs universitet

Now, could we proceed from here with first proving a

Lemma

In an isosceles trapezium (trapezoid here, in the US), that base is bigger which is incident to the smaller angles, and vice versa.

Now, since the bases are parallel, the sum of the interior angles incident to one sideline is 180°, meaning that the sum of the pair of the base angles adjacent to the smaller base is more than that. By the Fifth postulate, the side lines meet beyond the base with the larger pair of angles. This proves the lemma because it produces a pair of similar triangles of which one is inside the other so that the sides of the interior triangle are shorter than the corresponding sides of the exterior triangle and so are the bases.