## Triangle Areas / Steinhaus

**Click
here for a problem description**

Let us begin
by looking at a GSP sketch of the general problem.

To work with a GSP sketch of the problem click here
NOTE this sketch plots the ratio of the areas!

To watch a Quicktime movie illustrating the problem click here

Using GSP we can illustrate the relationship beween the ratio
of the blue area and the ratio of the red area and we get the
sketch above. The challenge is to derive this relationship by
some other means.

#### Derivation of Relationship

Finally we can show that area triangle AB''B = area triangle
BC''C = area triangle CA''A from which it follows that (note the
corresponding graph appears on the right):

**A special case** - the point on the edge
of the triangle trisects the edge.

There are two very convincing illustrations that the area of the
interior triangle is one-seventh of the exterior triangle.

**Illustration 1**: based on the illustration on the front
cover of *The Mathematics Teacher* (1992)

Click here
to look at a GSP animation.

Click here
to watch a movie of the animation.

**Illustration 2**: This is the illustration offered by Steinhauss
in his book *Mathematical Snapshots*.

NOTICE that the principle behind the two illustrations is really
the same.

**Return**