The congruum itself is four times the area of the same Pythagorean triangle.

Because is a square, and are the legs of another primitive Pythagorean triangle whose area .

It shows that, from any example of a Pythagorean triangle with square area, one can derive a smaller example.

It is also the length of the hypotenuse of four pythagorean triangles.

Among other topics, we will look at how we can dissect certain shapes into Pythagorean triangles.

Suppose there exists such a Pythagorean triangle.

There are no primitive Pythagorean triangles with integer altitude from the hypotenuse.

A Pythagorean triangle is right angled and Heronian.

Then it can be scaled down to give a primitive (i.e., with no common factors) Pythagorean triangle with the same property.

Thus "two" values, p and p/6, and the dimensions of the first two Pythagorean triangles are available from the solar diagram.

A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

Great stock, as you know, was placed in the almighty Pythagorean triangles; we were going to greet, across space, other civilizations - with Euclid's geometry.

There do not exist two Pythagorean triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.

Such a triangle is called decomposable, as dividing it into two similar smaller triangles with that altitude yields two more Pythagorean triangles.

During his lifetime, Fermat challenged several other mathematicians to prove the non-existence of a Pythagorean triangle with square area, but did not publish the proof himself.

For example a 5, 29, 30 Heronian triangle with area 72 cannot be constructed from two integer Pythagorean triangles since none of its altitudes are integers.

Because every congruum can be obtained (using the parameterized solution) as the area of a Pythagorean triangle, it follows that every congruum is congruent.

Thus, in short, each subtriangle is the result of scaling our primitive triangle by some positive integer scale factor and thus is still a Pythagorean triangle.

Suppose the sides of a Pythagorean triangle are , and , and suppose the angle between the leg and the hypotenuse is denoted as .

Such Pythagorean triangles are known as decomposable since they can be split along this altitude into two separate and smaller Pythagorean triangles.

Primitive Pythagorean triangles' sides can be written as , with a and b relatively prime and with a+b odd and hence y and z both odd.

Thus, any Pythagorean triangle with square area leads to a smaller Pythagorean triangle with square area, completing the proof.

All Greece needs to do is levy a small royalty on every comedy and tragedy, on every Pythagorean triangle, and on every piece of rhetoric and sophistry.

Geometrically, this means that it is not possible for the pair of legs of a Pythagorean triangle to be the leg and hypotenuse of another Pythagorean triangle.

Therefore, Fermat's proof that no Pythagorean triangle has a square area implies that this equation has no solution, and that this case of Fermat's last theorem is true.