Definition: Let $C_1$ and $C_2$ be circles in the plane with centers $(h_1,k_1),\ (h_2,k_2)$ and radii $r_1,\ r_2$ respectively and let $\alpha$ be an angle greater than 0 degrees and less or equal to 90 degrees. For each $0\le \theta \le 360$, let $P(\theta)$ be the intersection of the tangents to $C_1$ at $(h_1+r_1\,\cos \theta,k_1+r_1\,\sin \theta)$ and $C_2$ at $(h_2+r_2\,\cos (\alpha + \theta),k_2+r_2\,\sin (\alpha + \theta))$ respectively. The trace of the points $P(\theta)$ is a curve we call the bicycle curve (or bicycle) about $C_1$ and $C_2$ with angle $\alpha$.
Problem: Investigate the properties of bicycles.
Below is a Sagelet (Sage interact) which can be used to draw pictures of bicycles
for given of $C_1$ and $C_2$ respectively for given angle $\alpha$, and typical angle $\theta$. To see the bicycle, press the button. In this first sagelet, you can vary the centers and radii of the two circles and also the angle between the tangents.
Discussion:1) Try setting both radii $r_1$ and $r_2$ to 0. What theorem of circle geometry is brought to mind? Can you prove a generalization of that theorem to a theorem about bicycles?
2) When does the bicycle meet one of the circles? How many times does it meet one of the circles?
3) Does the bicycle always cross itself? How many times can this happen?
Start this one first.
Here is another sagelet where you can trace the track of the intersection of the tangents on the bicycle above. Make up some questions.