by jhncls on 10/29/24, 8:40 PM with 53 comments
by IsaacL on 11/1/24, 12:02 PM
https://cdn.britannica.com/43/70143-004-CCB17706/theorem-dem...
It is not immediately obvious why the area of the hypotenuse square should be equal to the sum of the areas of squares drawn on the other two sides of the triangle.
It is clear that the lengths of a, b and c are connected -- if we are given the length of any two of (a, b, c), and one angle, then the remaining side can only have one possible length.
So far, so simple; what is less clear is why the exact relationship for right triangles is c^2 = a^2 + b^2.
The other proofs demonstrate that the relationship holds, but give little insight.
The geometric proof linked above makes the relationship crystal-clear.
For any right triangle we can define a 'big square' with sides (a + b). The hypotenuse square is simply the area of the 'big square' with 4 copies of the original triangle removed.
Simple algebra then gives us the formula for the hypotenuse square:
The big square has area: (a+b)^2 = a^2 + 2ab + b^2
The original triangle has area: ab/2
1 big square minus four original triangles has area: (a+b)^2 - 4ab/2 = a^2 + b^2
Similarly, if you take the hypotenuse square, and subtract 4 copies of the original triangle, you get a square with sides (b - a). This is trivial to prove with algebra but the geometric visualisation is quite neat, and makes clear why the hypotenuse square must always equal the sum of the other two squares.
by tonystride on 10/30/24, 4:36 AM
I’m amazed by how many people I meet who don’t know about his contribution to the discovery and development of tonality! You mean the triangle guy invented music???
by sitkack on 10/30/24, 2:17 AM
https://www.youtube.com/watch?v=VHeWndnHuQs
What isn't stressed enough is that they both came up with their respective proofs independently.
by rhelz on 10/29/24, 8:55 PM
What is so counter-intuitive to me is that if the authors had wanted to earn $500 (or $250 after splitting it) they could have just got a job at McDonalds. They would have earned that money with far less time and effort.
I'm kinda glad that nobody pointed that out to them though :-)
But Prize-awards seems to put us into an entirely different economic frame. You can't say they did it just for the recognition, because if the prize wasn't there they wouldn't have bothered. But you also can't say that they did it for the money, because the money was ludicrously low--even when valued at the rate of unskilled labor.
by user070223 on 10/31/24, 9:38 PM
https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
by dr_dshiv on 10/31/24, 9:58 PM
by eointierney on 10/31/24, 11:56 PM
Apropos of nothing, just saying, and this thread is a great example.
I always want to read more books after a good dose of hacker news.
by eointierney on 10/31/24, 11:56 PM
Apropos of nothing, just saying, and this thread is a great example.
I always want to read more books after a good dose of hacker news.
by fn-mote on 10/30/24, 12:21 AM
However, I just cannot get excited about an article with proofs that:
(1) give a different name for methods that use sin(90)=1 vs only working with sine of an acute angle ("cyclometric" vs "trigonometric", ugh)
(2) use "high-powered" methods like convergence of infinite geometric series to prove the Pythagorean theorem
(3) apply the law of sines several times to produce the Pythagorean theorem
I just couldn't give it a chance. Give me a good old fashioned proof by a dissection diagram any day.
