Exercise 3: Why Garfield's Proof Shares the Core Dissection Strategy — Possible Solution
=============================================================================================

The core strategy behind dissection-style proofs of the Pythagorean
theorem is calculating the SAME total area two genuinely different
ways, then setting those two expressions equal to each other - since
both expressions describe the identical physical region, they must be
mathematically equal, and simplifying that equality reveals the
Pythagorean relationship.

A typical square-based dissection proof does this by taking a large
square built from the triangle's sides, computing its area as a single
large square, and separately computing it as the sum of smaller pieces
(the triangles plus inner square) arranged inside it - two expressions
for one shape's area.

Garfield's real trapezoid proof uses the exact same underlying logic,
just with a different starting shape. He builds a trapezoid instead of
a square, then computes ITS total area two different ways: once using
the standard trapezoid-area formula (treating it as one combined
shape), and once by adding up the areas of its three individual
pieces (the two right triangles and the square on the hypotenuse
between them). Setting these two area expressions equal and simplifying
produces the same Pythagorean relationship.

The shape chosen (square vs. trapezoid) is different, and the specific
pieces being summed are different, but the actual proof STRATEGY -
compute one region's area two independent ways and equate the results
- is identical. That's what makes Garfield's proof a genuine variation
within the same broader family of dissection proofs, not an entirely
unrelated method.

ANSWER: Garfield's proof shares the same core strategy as other
dissection-style proofs: computing one shape's total area two
independent ways (as a whole trapezoid via the standard formula, and as
the sum of its three constituent pieces) and setting the two results
equal. The specific shape (a trapezoid instead of a square) differs,
but the underlying "one area, two expressions" logic is identical.

WHY THIS WORKS AS AN ANSWER
------------------------------
This identifies the real shared logical strategy (double area
calculation set equal) underlying both Garfield's specific construction
and the general dissection-proof family, rather than treating them as
merely superficially similar.