There are thousands of different proofs of the Pythagorean theorem, and some of them are really cool. The purely trigonometric proof that was found by some high school students recently is a great one. However, I think the greatest proof of all is this little gem that has been attributed to Einstein [1].
Take any right triangle. You can divide it into two non-overlapping right triangles that are both similar to the original triangle by dropping a perpendicular from the right angle to the hypotenuse. To see that the triangles are similar, you just compare interior angles. (It's better to leave that as an exercise than to describe it in words, but in any case, this is a very commonly known construction.) The areas of the two small triangles add up to the area of the big triangle, but the two small triangles have the two legs of the big triangle as their respective hypotenuses. Because area scales as the square of the similarity ratio (which I think is intuitively obvious), it follows that the squares of the legs' lengths must add up to the square of the hypotenuse's length, QED.
It's really a perfect proof: it's simple, intuitive, as direct as possible, and it's pretty much impossible to forget.
unfortunately doesn't work for me because of difficulty visualizing things, so I suppose there are probably a good number of people with the same problem.
So I guess for one particular subset of the population it is difficult, impossible to understand, and because it cannot be understood it will not be remembered.
Not complaining just noting the amusing thing that different explanations may have all sorts of problems with it.
Although if there was a video of it I guess I would understand it then. Not sure if everyone with visualization issues would though.
WCSTombs•1h ago
Take any right triangle. You can divide it into two non-overlapping right triangles that are both similar to the original triangle by dropping a perpendicular from the right angle to the hypotenuse. To see that the triangles are similar, you just compare interior angles. (It's better to leave that as an exercise than to describe it in words, but in any case, this is a very commonly known construction.) The areas of the two small triangles add up to the area of the big triangle, but the two small triangles have the two legs of the big triangle as their respective hypotenuses. Because area scales as the square of the similarity ratio (which I think is intuitively obvious), it follows that the squares of the legs' lengths must add up to the square of the hypotenuse's length, QED.
It's really a perfect proof: it's simple, intuitive, as direct as possible, and it's pretty much impossible to forget.
[1] https://paradise.caltech.edu/ist4/lectures/Einstein%E2%80%99...
bryanrasmussen•6m ago
So I guess for one particular subset of the population it is difficult, impossible to understand, and because it cannot be understood it will not be remembered.
Not complaining just noting the amusing thing that different explanations may have all sorts of problems with it.
Although if there was a video of it I guess I would understand it then. Not sure if everyone with visualization issues would though.