Squish

If one accepts the circle as point of reference, then any triangle
within one half circle will have a right angle at its apex.
Indeed, one is creating a half box when the entire circle is considered.
We are just playing squishy boxes.

And if we assign our right angle 90 degrees, then any triangle will contain
180 (because any box has four).

The area of our right triangle will be one-half that of its box.

The maximal inscribed box we can create is a square.
Its area will be one-half that of an outer box.

Squaring both sides will yield two inner boxes; thus, the area
of the outer box.

In point of fact, the circle is an orientation mechanism. Side b approaches
the value of the diameter in a squishier triangle, and side a approaches 0.
The square of the sides will always be equal to that of the hypotenuse.