Pythagoras of Samos was a Greek philosopher who lived around 530 BC,
mostly in the Greek colony of Crotona in southern Italy. According to tradition
he was the first to prove the assertion (theorem) which today bears his
name:
A right angle can be defined here as the angle formed when two straight lines cross each other in such a way that all 4 angles produced are equal. The theorem also works the other way around: if the lengths of the three sides ( a,b,c) of a triangle satisfy the above relation, then the
angle between sides a and b must be of 90 degrees.
For instance, a triangle with sides a = 9 + 16 = 25 = c Ancient Egyptian builders may have known the (3,4,5) triangle and used it (with measured rods or strings) to construct right angles; even today builders may still nail together boards of those lengths to help align a corner.
means 2ab2 times times
a).
For example
b- 15
^{2} = (10 + 5)^{2}
= 10 ^{2} + (2)(10)(5) + 5^{2}
= 100 + 100 + 25 = 225
- 5
^{2} = (10 - 5)^{2}
= 10 ^{2} - (2)(10)(5) + 5^{2}
= 100 - 100 + 25 = 25 ab.
A diagonal cut divides it into two right-angled triangles with short sides
a
and b, and the area of such a triangle is therefore (1/2)
ab.
Now look at the square on the left constructed out of four ( However, the square can also be divided into four (
Using the identity for (
Subtract 2
The same result can also be shown using a different square, of area
a-b). We get
Q.E.D. stands for "quod erat demonstrandum," Latin for "which was to
be demonstrated," and in traditional geometry books those letters mark
the end of a proof. The importance of the work of Pythagoras and of later
masters of Greek geometry (especially Euclid) was not only in |