Equations of two circles coincide (Soft 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?

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