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.