RE: Proof by Rearrangement of the Pythagorean Theorem
November 22, 2018 at 6:44 pm
(This post was last modified: November 22, 2018 at 6:49 pm by polymath257.)
I'd also point out there are other implicit assumptions here. For example, how is the square of side length a+b constructed? Well, we can take a line segment of length a+b, say AB. Do perpendiculars from both ends, giving sides AC and BD, each of length a+b.
iI is an *assumption* that the line CD is perpendicular to both AC and BD and is of length a+b. In non-Euclidean geometry, this line is NOT perpendicular to either AC or BD (the angle is *less* than a right angle) and its length is *more* than a+b. There *are* no squares in Lobachevskian geometry with four equal sides and right angles. They simply don't exist.
So even the original figures have hidden assumptions. To be perfectly rigorous, all these details would need to be addressed.
iI is an *assumption* that the line CD is perpendicular to both AC and BD and is of length a+b. In non-Euclidean geometry, this line is NOT perpendicular to either AC or BD (the angle is *less* than a right angle) and its length is *more* than a+b. There *are* no squares in Lobachevskian geometry with four equal sides and right angles. They simply don't exist.
So even the original figures have hidden assumptions. To be perfectly rigorous, all these details would need to be addressed.
