Comment by codethief
> So just choose units where our triangle itself with a given hypotenuse H has area H^2 units.
This is not at all trivial. You're claiming you can choose units in such a way (reusing my notation from before) that simultaneously
A = a², B = b², C = c².
Intuitively, you can do that precisely because the triangles are similar and area is quadratic in the similarity ratio. But there is definitely some algebra behind that.
To be clear, I'm just claiming that we can choose a specific area unit, and the three equations you wrote are then obvious consequences of that. It's true, you do need to assume area scales as the square of length, but IMO that's a pretty fundamental fact, and I think that's intuitive for many others. But as always, YMMV.