Explorations in Geometry

with the

Geometer’s Sketchpad

Exploration 19.

In this exploration, you will use what you learned about tangent circles to discover a relationship between the radii of three circles which are mutually tangent to each other and an external line. This proof was drawn up on a wooden block and hung in a temple in Gunma prefecture of Japan in the early 1800's . This practice was called Sangaku. You can learn more about the Japanese temple geometry by reading the article in the May, 1998 issue of Scientific American.

- Draw a line and create three circles that are tangent to the line and each other.

2) Use what you learned about the lengths of the external segments of the common tangent to write an equation involving all three tangents.

3) Simplify the expression for the radius of the smaller circle in terms of the other two.

4) Try several different sizes of circle groups and look for a relationship in the angles formed in the triangle joining the three centers.

5) What else can you discover about this figure.

6) Write a brief paragraph and describe the relationships you have observed in this exploration.

7) Read the issue of Scientific American described above and try to prove one of the other Sangaku problems described there.