**Fun
with Circles**

**By
**

**Jeffrey
R. Frye**

**Purpose: **The focus of this set
of problems is to provide some challenges to students that will allow them to
have some fun while attempting to solve the puzzle presented by the construction
of the design. These explorations are
directed at having some fun by first exploring the **how to** construct the design, and then discovering the mathematics
that are associated with completing these constructions of different kinds of
circle designs. Some basic mathematical analysis is presented leaving the more
in depth mathematical calculations to the reader or student that has an
interest in going beyond the construction itself.

**I. Circle Window**

There are several items in the first construction
that lend themselves to further extension and exploration. The “circle window” offers some interesting
challenges for the student to be able to **construct
**this window and not just draw it. By
starting with the window and asking the student to explore and find the
solution, the challenge is begun. One
starting point is to construct three congruent central angles in a circle as
shown below.

From the “circle window” design above, it can be
seen that three circles are constructed inside the larger circle, and that
these circles are tangent to the larger circle, as well as being tangent to
each other. How is this
accomplished? There are probably several
methods that can be used, but the one that is used in this presentation
incorporated the properties of the triangle incenter. By constructing three congruent triangles and
finding the incenter of each, the three tangent circles inside the larger
circle can be constructed. This is one
of the hints that will help in constructing the “circle window”. See the construction below.

Now the remainder of the construction requires that
arcs be constructed and combined in a way that the design is complete. Click here
to explore the “circle window” construction.

**II.
Bradford Wool Exchange Window**

This design incorporates several relationships into
the construction. A starting point
involves dividing the circle into seven equal arcs and then constructing a
heptagon. There are several methods that
can be employed to make this construction.
Check them out here. This construction used method 1, but the
other methods are interesting, too. With the heptagon constructed, the location of
the points of tangency for the seven circles with the larger and smaller
circles can be located. How are the
radii of the small circle and the seven circles determined? The Star of David is then constructed inside
the smaller circle. Enjoy working with this one. If you decide that you need help,
this will provide some answers and direction.
The GSP illustration is a beginning
point for this construction; much work is still needed.

**III.
Double Vesica**

This construction uses the relationships found in one
of the “special right triangles”, the 30-60-90 right triangle. The intersection of arcs made from this
radius relationship, results in the above design. The mathematics has been left to the explorer
on this design. Click here to explore using GSP.

**IV.
Stacking Circles**

For this construction, the circles are tangent and the
radius of the next lower group of circles is one-half of the radius of the circle
above it. By constructing segments that
connect the centers of the circles, similar triangles can be constructed. To explore how the tangent circles are
constructed and to view the similar triangles, click here.