Assignment 7 - Problem #1 for Kanita Ducloux

Tangent Circles

Prepare a retrospective summary of the following problem:

Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.

Frustrating! Hair pulling! Problematic!

That's how I would describe this problem. I took notes on the lesson and even asked Dr. Wilson for help. But I was still lost. So, after much prayer, meditation, studying my notes, and trying to recall my discussion with Dr. Wilson, I was finally able to do something similar to the what Dr. Wilson illustrated.

1. Draw a circle within a circle and a point (B) on one of the circles.

2. Find the radius of the small circle and construct a circle of the same radius with point of trangency (B) being the center of the new circle.

3. Construct a line through the center of the large cirlce and the point of tangency (B).

4. Construct a segment connecting K to the center of the original small circle and find the midpoint (E).

5. Construct the perpendicular bisector of the segment. The intersection of the perpendicular bisector and the line connecting B to the center the of the original small circle is point F.

6. Finally, construct the tangent circle with center F and point of tangency B.

After all of that, I thought I was finished until I discovered that there was another such circle tangent to the two given circles. How do I find it? I should be able to use what I've already done. There should be a relationship between the two circles. Let's see what I found.

In step 4, instead of connecting K to the center of the original small circle, connect to the second point of intersection (K2).

5B. Construct the perpendicular bisector of the segment. The intersection of the perpendicular bisector and the line connecting B to the center the of the original small circle is point F.

6. Finally, construct the tangent circle with center F2 and point of tangency B.

Let's look at the picture without all of the lines. The two green circles are the tangent circles at Point B.

I was glad when I finallly finished this problem.