TITLE: Maximum number of tangents to two circles in affine geometry
QUESTION [3 upvotes]: How would one prove that the maximum number of tangents to two circles is 4, without recurring to the equations of the circles? I have found several ways of determining them (most of them using Calculus), but not some proof.

REPLY [1 votes]: Let $ax+by+c=0$ be an equation of the tangent to two circles with centers $(x_1,y_1)$ and $(x_2,y_2)$ and radii $R_1$ and $R_2$ respectively. 
Thus, $$\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}=R_1$$ and
$$\frac{|ax_2+by_2+c|}{\sqrt{a^2+b^2}}=R_2.$$
I think now we can see it.
{"set_name": "stack_exchange", "score": 3, "question_id": 3109126}