My colleague Ricardo Pérez-Marco showed me a very cute proof of Pythagoras’ theorem, which I thought I would share here; it’s not particularly earth-shattering, but it is perhaps the most intuitive proof of the theorem that I have seen yet.

In the above diagram, a, b, c are the lengths BC, CA, and AB of the right-angled triangle ACB, while x and y are the areas of the right-angled triangles CDB and ADC respectively. Thus the whole triangle ACB has area x+y.

Now observe that the right-angled triangles CDB, ADC, and ACB are all similar (because of all the common angles), and thus their areas are proportional to the square of their respective hypotenuses. In other words, (x,y,x+y) is proportional to . Pythagoras’ theorem follows.

Here is a more “modern” way to look at Pythagoras’ theorem. The statement is equivalent to the assertion that the matrices and have the same determinant. But it is easy to see geometrically that the linear transformations associated to these matrices differ by a rotation, and the claim follows.

Homework: why are the above two proofs essentially the same proof?

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6 January, 2009 at 9:59 pm

lalit warkdewatch any one of the links below to get the best explanation for pythagoras theorem

http://www.authorstream.com/Presentation/lalitwarkde-85464-pythagoras-theorem-mathematics-lalit-education-ppt-powerpoint/

7 March, 2011 at 11:49 pm

Nilotpal SinhaDear Prof. Tao

I think there is a typographical error in the line “while x and y are the areas of the right-angled triangles ADC and CDB respectively.” As per the diagram, x and y are the areas of CDB and ADC respectively instead of the reverse order.

[Corrected, thanks – T.]Regarding the Pythagoras theorem, one of the most beautiful and non-trivial one is due to Dijkstra. You probably might already know it but I would like to post it just in case.

Theorem (Djikstra). If a, b, c are the sides a triangle and A, B and C are the respective opposite angles then sign(A + B – C) = sign(a² + b² – c²).

In a right triangle, A + B = 90 = C. Hence LHS is zero and this implies a² + b² = c². We have the Pythagoras Theorem as a special case of this simple looking theorem.8 March, 2011 at 1:55 am

Nilotpal SinhaDear Prof. Tao,

In his monumental book Elements, Euclid has proved the following theorem.

Theorem. (Euclid). If one erects similar figures on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.

The beauty of Euclid’s theorem is that there is no restriction on the shape of figures erected along the sides of a right triangle as long as they are similar. This proof by Pérez-Marco is a special case of Euclid’s theorem where we have erected similar right triangles on the sides of a right triangle.

4 September, 2011 at 5:37 am

Luqing YePythagoras’ theorem is so basic.So its proof,the simpler,the better :)

4 September, 2011 at 5:40 am

Luqing YeAnd i think one basic proof is enough(like the one you mentioned), other “brilliant”,”amazing” proofs are not necesssary.

6 September, 2011 at 2:02 am

AnonymousThese are excellents mathematicans,Pythagoras,Terence Tao, and above all.

8 September, 2011 at 9:11 pm

Recent reading – Pythagoras’ Theorem « Linear Algebra[…] I read a pretty old post of Prof Tao on Pythagoras’ Theorem. All of us know what Pythagoras’ theorem says, and the 1st part of Tao ‘s post gives a […]

17 May, 2013 at 7:48 am

Richard PalaisAs has been mentioned, the proof you give Terry (which is definitely my own favorite), is often attributed to Einstein. In fact, Einstein attributes it to himself ! In his autobiography he says he discovered it at age twelve when his uncle told him about the theorem.

29 May, 2013 at 10:21 pm

Mustafa SaidRecently, I have found a way to generate a family of normal matrices with integer entries using Pythagorean triples. The construction may be new, as I have not been able to find it in the literature. If anyone is interested, I can send you my note.

26 August, 2013 at 5:12 am

Colin McLartyThe earliest recorded use of the Pythagorean theorem in a proof to my knowledge (by far not the earliest hint that people knew the theorem) suggests this is the proof Hippocrates of Chios used in the 5th century BCE. He uses the theorem not for squares but for segments of circles cut off by the sides of an isoceles right triangle, to prove his famous theorem on areas of lunes, which suggests he saw it as bascially a theorem on scaling arbitrary figures. See http://en.wikipedia.org/wiki/Lune_of_Hippocrates

26 August, 2013 at 8:16 am

sushma sharmaA pythagoras triplet represents the lengths of the sides of a right triangle where all three sides have integer length..!!

14 November, 2013 at 3:04 pm

Shay Ben MosheI find this proof wonderful, however I think that using the fact that the determinant of a rotation matrix is 1 is circular, because we use the identity $sin^2(x)+cos^2(x)=1$, which follows from the Pythagoras’ theorem.

A solution might be stating that this matrix take one orthonormal basis to another, and therefore the matrix is orthonormal, and thus its determinant is 1.

21 November, 2013 at 2:09 am

Nice proofs of Pythagoras’ theorem | mathbeauty[…] Pythagoras’ theorem | What’s new. […]

19 January, 2015 at 12:17 am

Pythagoras’ theorem | Cassandra Lee Yieng[…] So we have Pythagoras’ Theorem: in a right-angled triangle, adding up the squares of the two shortest sides gives the square of the longest one. Terence Tao posted an intuitive proof here. […]

3 March, 2015 at 10:40 pm

CliffInteresting extensions of Pythagoras theorem.

Bet you didn’t know this about Pythagoras.

4 March, 2015 at 3:23 am

AnonymousTo find all Pythagorean triples with consecutive , let and observe that

One can verify that the solution of this (Pell’s) equation (for the n-th such triple) is given by

Which implies (for the n-th such triple) the simple formula

Showing asymptotically geometric growth of the solutions with (asymptotic) ratio .

The first few solutions are:

with corresponding Pythagorean triples solutions

30 April, 2015 at 7:11 pm

A more motivated proof of the Pythagorean theorem | Daniel McLaury's Mathematical Diary[…] Pythagoras’s theorem | What’s New […]

14 July, 2016 at 12:13 am

AnonymousI wish you had explained it well instead of being elusive.

9 January, 2017 at 9:24 am

AnonymousThe differentials in Pi and Pythagoras are all same only look different…, the order of all is founded in One…..

13 September, 2020 at 11:23 pm

KevinHere is an interesting proof involving dividing a triangle into infinitely many smaller triangles and using infinite geometric series:

https://ibtaskmaker.com/maker.php?q1=618

14 September, 2020 at 10:48 am

AnonymousIs the Euclidean norm in R^2 the only norm which is invariant under a positive dimensional group of linear transformation?

16 September, 2020 at 6:32 am

Terence TaoUp to a linear change of coordinates, yes (this follows from the classification of one-dimensional Lie subgroups of , which arise from applying the exponential to one-dimensional Lie subalgebras of and ruling out those groups that are non-compact; one can also use Weyl’s unitary trick). If one allows degenerate metrics or pseudometrics then one also has the Galilean semimetric and the Lorentz pseudometrics , and linear transforms thereof (cf. the classification of elements of into elliptic, parabolic, and hyperbolic elements).

4 December, 2021 at 6:00 pm

Het verhaal achter: de Pythagoraskalender (1) – Bollebus[…] Tao, Pythagoras’ Theorem. Blog, 2007. […]