Equations of two circles coincide (Soft question)

Question

As the title suggests, I would like to ask if I have two equations of circle that coincide, are they considered tangent to one another?

I am currently working on a proof of Feuerbach's Theorem (1822) which states that the nine-point circle of a triangle is tangent to the incircle and the point of tangency of these two circles is called the Feuerbach point of the triangle.

I have stumbled upon a case where the triangle is equilateral and the incircle and the nine-point circle coincide. In this case, can we consider these circles as tangent to one another?

Answer 1

No, two circles are tangent only if they meet at exactly one point.

Answer 2

Consider Euclidean three dimensional space with a distinguished oriented plane $P$. Given any point $Q$ in space, perpendicularly project that point onto $P$ at point $O_Q$ and draw a circle $C_Q$ in $P$ with center at $O_Q$ and radius being the distance from $Q$ to $O_Q$. The circle $C_Q$ will be oriented positively if $Q$ is on the positive side of $P$. This sets up a bijection between points in space and oriented circles in $P$.

Given any point $Q$ and its corresponding circle $C_Q$, then any other circle tangent to $C_Q$ and oriented in the same dirction will correspond to a point on the double circular cone with vertex at $Q$ and intersecting $P$ at $C_Q$. All the points on the double cone represent circles tangent to $C_Q$. The point $Q$ itself is the vertex of the cone and hence, it seems that the circle $C_Q$ is tangent to itself. This may or may not make sense depending on the context. If you need to use the tangency point, then it is not unique in the case of a circle being tangent to itself.

This seems similar to the case of the intersection of two lines in a plane. If they are parallel, then it would seem that there is no intersection point. However, if the "line at infinity" is introduced, then any two different lines intersect at only one point, but that point may be a point on the line at infinity. Even in this case, is a line tangent with itself? If we require one and only one point of intersection, then no, but if we allow at least one point in common, then yes. This is dual to two distinct points determine a unique line. If the two points are the same, they do not determine a unique line. This is very similar to division by zero.

The Wikipedia article Feuerbach point gives the trilinear coordinates for the Feuerbach point as one minus the cosine of angle differences. In an equilateral triangle all of the angles are equal and hence the trilinear coordinates are $\,0:0:0\,$ which does not define a point. The Feuerbach point is not defined in this case. This is consistent with a circle not being tangent to itself. The Wikipedia article does not currently mention the special case of an equilateral triangle not having a Feuerbach point. Sometimes special cases are ignored.