I was playing with two rings today, and put one inside the other. It made me think about how many points of a circle touch another circle that is smaller, given that the farthest point of the smaller circle in a given direction touches the farthest point of the bigger circle in the same direction.
Given a tangent line to a circle, the line is, well, tangent. Let's say you have the circle described by:
and the line described by:
Obviously, those are tangent. Now, given the same circle:
but instead of a line, another circle:
$(x-1)^2+y^2 = 2^2$
Isn't it feasible to say that larger circle should have more than one intersection with the smaller circle, because given:
$L=$ distance between line and horizontally-aligned portion of smaller circle.
$C=$ distance between left half of bigger circle and horizontally-aligned portion of smaller circle.
$C$ is always less than $L$ except when $L=0$. Why can't a value of $C$ that is related to a value $L$ (related in the sense that they represent the distance from their assigned object to the smaller circle on the some vertical level) when $L$ is incredibly small be equal to $0$?
If I were to guess without doing any math, I would say $C=Lm$ where $m$ is some function or multiple that is less than zero, and changes the further the vertical distance from the x-axis.