Garfield's proof of the Pythagorean Theorem

Einstein-style similar-triangle proof & area scaling

• A popular proof (attributed to Einstein) splits a right triangle into two smaller similar right triangles by dropping a perpendicular to the hypotenuse.
• Using similarity, the legs of the original become hypotenuses of the smaller triangles; areas add, and since area scales with the square of a length scale factor, one gets (a^2 + b^2 = c^2).
• Some readers find this very elegant, simple, and unforgettable; others struggle to visualize it from text and note that, in practice, a diagram is essential.
• A recurring debate: how “obvious” is it that area scales with the square of a length (and that the proportionality constant is the same for similar triangles)? Several comments supply justifications:
  • Via “base × height / 2” and scaling both base and height.
  • Via similar figures and unit choices for area.
  • Via informal dimensional arguments (area is 2D, length is 1D).

Linear algebra / determinant-based proofs

• A linked writeup using matrices and rotations drew criticism: the step “these differ by a rotation” feels like it already assumes what’s being proved.
• Discussion centers on whether one can show a rotation matrix has determinant 1 without smuggling in the Pythagorean identity (e.g., via (\cos^2 + \sin^2 = 1)).
• Some suggest defining determinant via area or vice versa, but there is concern about hidden reliance on Pythagoras in such constructions.

Garfield’s trapezoid proof and classic square proofs

• Several note Garfield’s trapezoid proof is essentially “half” of the classic square-with-four-triangles proof; pairing two trapezoids reconstructs the familiar ((a+b)^2 = c^2 + 2ab) argument.
• Some find Garfield’s version needlessly complicated (requiring the trapezoid-area formula) compared with dropping an altitude or using the standard square construction; others value that it uses very basic area facts.
• Another commenter points to a related similar-triangle proof that uses only elementary algebra and may be easier to follow.

Intuition, explanation style, and the “magic” of Pythagoras

• Several people say that even with many proofs—geometric, trigonometric, linear-algebraic—the theorem still feels “magical”: the squared perpendicular distances summing to the squared straight-line distance.
• There’s meta-discussion about mathematicians leaving “obvious” steps to the reader and how this can alienate those without strong geometric intuition.
• One analogy compares this to a world where music is only audible to dogs: experts are working with intuitions most people can’t directly “hear.”

Arbitrary shapes, non-Euclidean twists, and related curiosities

• The idea that Pythagoras works with any congruent shapes on the sides (even a face or arbitrary polygon) is seen as both powerful and still somewhat inexplicable at an intuitive level.
• Some note that all of these arguments assume Euclidean geometry; in curved or other metric spaces, Pythagoras changes form or fails.
• Related links surface: the Pythagoras tree fractal, non-Euclidean variants (spherical, hyperbolic), and integer hypotenuse sequences.

History, attribution, and Garfield as president

• One thread emphasizes that the theorem predates Greek sources, with evidence of its use or statement in ancient Indian, Babylonian, and Egyptian traditions; attribution to Pythagoras is historically murky.
• Another subthread discusses Garfield himself: his intellectual breadth, early death by assassination, civil-service legacy, and a recent dramatized miniseries.
• Several commenters wistfully contrast his mathematical ability with modern political figures, spinning off into light political and sci-fi jokes.

Humor and pop-culture tangents

• Many clicked expecting the cartoon cat and lasagna, or pizza-slice triangle proofs, and express mock disappointment.
• Other light references: a novel proof via cake in a science-fiction novel, a TV host bungling the theorem, and conspiratorial jokes about “forbidden triangle knowledge.”