Triangles Inscribed Inside Triangles

Given any triangle, construct a midpoint on each side. Connect each consecutive midpoint with a segment so that four small triangles appear. Explain the relationship between the four triangles. How does the area of one of the smaller figures compare to the area of the original triangle? What types of quadrilaterals are formed inside the largest triangle? Explain how you found your answers.


During an iteration, smaller triangles are created by connecting the midpoints of a larger triangle. In the diagram above, you started with one triangle, and then used the first iteration to create four more triangles by connecting the midpoints. The middle white triangle is called a medial triangle. How many more triangles will there be in the second iteration if you connect the midpoints of the smaller triangles (except for the medial triangle)?

How many more triangles will be added during the eighth iteration? Explain your results.