century than a Euclidean proof from 300 BC.
This is a proof ‘without words’. In the figure the square with side a + b can be
divided in two different ways.
Since the four equal triangles (shaded dark) are common to both squares we
can remove them and still have equality of area. If we look at the areas of the
remaining shapes, out springs the familiar expression.
a^2 + b^2 = c^2
The Euler line
Hundreds of propositions about triangles are possible. First, let’s think about
the midpoints of the sides. In any triangle ABC we mark the midpoints D, E, F of
its sides. Join B to F and C to D and mark the point where they cross as G. Now
join A to E. Does this line also pass through G? It is not obvious that it should
necessarily without further reasoning. It fact it does and the point G is called the
‘centroid’ of the triangle. This is the centre of gravity of the triangle.
There are literally hundreds of different ‘centres’ connected with a triangle.
Another one is the point H where the altitudes (the lines drawn from a vertex
perpendicular to a base – shown as dotted lines in the figure on page 86) meet.
This is called the ‘orthocentre’. There is also another centre called the
‘circumcentre’ O where each of the lines (known as ‘perpendiculars’) at D, E and F
meet (not shown). This is the centre of the circle which can be drawn through A,
B and C.