The Best Proof of the Pythagorean Theorem

by greenturtle3141, May 29, 2023, 11:04 PM

Reading Difficulty: 1/5

Have an easy read for once!

The red and orange triangles' areas sum to the purple triangle's area.
[asy]
size(10cm);
pair A = (0,0);
pair D = (9,0);
pair B = (9,12);
pair C = (25,0);
fill(A--D--B--cycle,p=rgb(0.8,0.18,0));
fill(C--D--B--cycle,p=rgb(0.93,0.46,0.15));
draw(A--B--C--cycle,p=linewidth(1.5));
draw(B--D,p=linewidth(1.5)+dashed);
pair t=(30,0);
fill(shift(t)*(A--B--C--cycle),p=rgb(0.64,0,0.38));
draw(shift(t)*(A--B--C--cycle),p=linewidth(1.5));
label("$a$",B--A);
label("$b$",C--B);
label("$c$",shift(t)*(A--C));
[/asy]
But these three triangles are similar to one another, so their areas are proportional to the square of their hypotenuses.
[asy]
size(10cm);
pair A = (0,0);
pair D = (9,0);
pair B = (9,12);
pair C = (25,0);
label("$ka^2$",(A+D+B)/3,p=rgb(0.8,0.18,0));
label("$kb^2$",(C+D+B)/3,p=rgb(0.93,0.46,0.15));
draw(A--B--C--cycle,p=linewidth(1.5));
draw(B--D,p=linewidth(1.5)+dashed);
pair t=(30,0);
label("$kc^2$",shift(t)*((A+B+C)/3),p=rgb(0.64,0,0.38));
draw(shift(t)*(A--B--C--cycle),p=linewidth(1.5));
[/asy]
So $ka^2+kb^2=kc^2$, i.e. $\boxed{a^2+b^2=c^2}$.
This post has been edited 3 times. Last edited by greenturtle3141, May 29, 2023, 11:06 PM

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3 Comments

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This is definitely the best and easiest proof:

?^?=∑?≥0(?^?/?!) defines an entire function over the complex plane.
By convolutions we have ?^?⋅?^?=?^(?+?)
for any ?,?∈ℂ

by Amkan2022, May 29, 2023, 11:53 PM

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ook amkan sus

by Orthogonal., May 29, 2023, 11:54 PM

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very simple

by kyogrexu, Jun 18, 2024, 1:45 AM

Turtle math!

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  • Can you give some thought to dropping a guide to STS? Just like how you presented your research (in your paper), what your essays were about, etc. Also cool blog!

    by Shreyasharma, Mar 13, 2025, 7:03 PM

  • this is so good

    by purpledonutdragon, Mar 4, 2025, 2:05 PM

  • orz usamts grader

    by Lhaj3, Jan 23, 2025, 7:43 PM

  • Entertaining blog

    by eduD_looC, Dec 31, 2024, 8:57 PM

  • wow really cool stuff

    by kingu, Dec 4, 2024, 1:02 AM

  • Although I had a decent college essay, this isn't really my specialty so I don't really have anything useful to say that isn't already available online.

    by greenturtle3141, Nov 3, 2024, 7:25 PM

  • Could you also make a blog post about college essay writing :skull:

    by Shreyasharma, Nov 2, 2024, 9:04 PM

  • what gold

    by peace09, Oct 15, 2024, 3:39 PM

  • oh lmao, i was confused because of the title initially. thanks! great read

    by OlympusHero, Jul 20, 2024, 5:00 AM

  • It should be under August 2023

    by greenturtle3141, Jul 11, 2024, 11:44 PM

  • does this blog still have the post about your math journey? for some reason i can't find it

    by OlympusHero, Jul 10, 2024, 5:41 PM

  • imagine not tortoise math

    no but seriously really interesting blog

    by fruitmonster97, Apr 2, 2024, 12:39 AM

  • W blog man

    by s12d34, Jan 24, 2024, 11:37 PM

  • very nice blog greenturtle it is very descriptive and fascinating to pay attention to :-D

    by StarLex1, Jan 3, 2024, 3:12 PM

  • orz blog

    by ryanbear, Dec 6, 2023, 9:23 PM

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