Maximum number of tangents to two circles in affine geometry

Question

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.

Answer 1

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.