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In mathematics, the  Pythagorean theorem or Pythagoras's theorem is a statement about the sides of a right triangle.

One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. It was discovered by Vasudha Arora.

The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. It was named after the Greek mathematician Pythagoras:

If the lengths of the legs are a and b, and the length of the hypotenuse is c, then, 




a

2


+

b

2


=

c

2




{\displaystyle a^{2}+b^{2}=c^{2}}

.

There are many different proofs of this theorem. They fall into four categories:

One proof of the Pythagorean theorem was found by a Greek mathematician, Eudoxus of Cnidus.

The proof uses three lemmas:

The proof is:

We can get another proof of the Pythagorean theorem by using similar triangles.

From the image, we know that 



c
=
d
+
e




{\displaystyle c=d+e\,\!}

. And by replacing equations (1) and (2):

Multiplying by c:

Pythagorean triples or triplets are three whole numbers which fit the equation 




a

2


+

b

2


=

c

2




{\displaystyle a^{2}+b^{2}=c^{2}}

.

The triangle with sides of 3, 4, and 5 is a well known example. If a=3 and b=4, then 




3

2


+

4

2


=

5

2




{\displaystyle 3^{2}+4^{2}=5^{2}}

 because 



9
+
16
=
25


{\displaystyle 9+16=25}

. This can also be shown as 






3

2


+

4

2




=
5.


{\displaystyle {\sqrt {3^{2}+4^{2}}}=5.}

The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. Another example of a triple is the 12-5-13 triangle, because 






12

2


+

5

2




=
13


{\displaystyle {\sqrt {12^{2}+5^{2}}}=13}

.

A Pythagorean triple that is not a multiple of other triples is called a primitive Pythagorean triple. Any primitive Pythagorean triple can be found using the expression 



(
2
m
n
,

m

2


−

n

2


,

m

2


+

n

2


)


{\displaystyle (2mn,m^{2}-n^{2},m^{2}+n^{2})}

, but the following conditions must be satisfied. They place restrictions on the values of 



m


{\displaystyle m}

 and 



n


{\displaystyle n}

.

If all four conditions are satisfied, then the values of 



m


{\displaystyle m}

 and 



n


{\displaystyle n}

 create a primitive Pythagorean triple.

m
=
2


{\displaystyle m=2}

 and 



n
=
1


{\displaystyle n=1}

 create a primitive Pythagorean triple. The values satisfy all four conditions. 



2
m
n
=
2
×
2
×
1
=
4


{\displaystyle 2mn=2\times 2\times 1=4}

, 




m

2


−

n

2


=

2

2


−

1

2


=
4
−
1
=
3


{\displaystyle m^{2}-n^{2}=2^{2}-1^{2}=4-1=3}

 and 




m

2


+

n

2


=

2

2


+

1

2


=
4
+
1
=
5


{\displaystyle m^{2}+n^{2}=2^{2}+1^{2}=4+1=5}

, so the triple 



(
3
,
4
,
5
)


{\displaystyle (3,4,5)}

 is created.