In triangle , segment from vertex meets opposite side at a point with the length of 1/3 the length of . That is, point is located 1/3 of the way along segment from *C*. Similarly, points and are located at 1/3 marks along the other two sides of triangle . The intersections of these segments are the vertices of another triangle in the interior of triangle .

What is the relationship between the areas of triangles and ?

Source: NCTM Problem to Ponder 11/16/2010

**SOLUTION
Part I**

because

because

because

Looking at the figure we notice that, when we are adding areas, the area of triangle is completely left out and that the areas of the three shaded triangles , and are each counted twice. In order for Eq. (1) to be satisfied, we conclude that

**Part II
**Draw parallel to , parallel to and parallel to .

**Triangle is similar to triangle with a scale factor of**

Triangle is similar to triangle with a scale factor of

Triangle is similar to triangle with a scale factor of

By property of areas of similar triangles

By property of proportion

Substitute the value of the numerator from Eq. (2)

**Part III**

Draw a parallel line from vertex to and a second parallel line from vertex to . The two parallel lines intersect at point .

Draw a parallel line from vertex to and a second parallel line from vertex to . The two parallel lines intersect at point .

Draw a parallel line from vertex to and a second parallel line from vertex to . The two parallel lines intersect at point .

The constructions create congruent parallelograms and triangles (in the interior of parallelograms) that have equal area because they are halves of congruent parallelograms. For example,

Let

We want to show that

Triangle triangle by ASA

alternate interior angles

alternate exterior angles

Similarly,

Triangle triangle

Triangel triangle

We have

Adding equations (4), (5), and (6)

by Eq. (2)

**Summary**

We have decomposed the entire triangle into 7 distinct and contiguous triangles the areas of which are related as follows:

by Eq. (3)

by Eq. (7)

The ratio is .

**Answer**: .