# Maximum number of tangents to two circles in affine geometry

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.

• You have to be a little careful, because this isn't true if the two circles coincide - then every tangent to one is tangent to the other, but that just requires you to mention two distinct circles. Feb 11, 2019 at 19:05
• The argument here showing how to construct the tangents may offer a clue to a proof. cut-the-knot.org/Curriculum/Geometry/GeoGebra/… Feb 11, 2019 at 19:23
• @Mark Bennet He wants a maximal number of tangents. Feb 11, 2019 at 19:25
• @J Dionisio For non intersecting circles when one is not contained inside the other (external non-intersection circles) there are an infinite number.of tangents. If the point is given / or is fixed outside both of them then only there can be four tangents. So your question needs to be more specific.Depending upon $(d-R_1-R_2 )$ center distance =d their number varies. Feb 11, 2019 at 19:55
• @Narasimham an infinite number of lines who are tangents to two circles at the same time? Maybe I'm not understanding what you wrote, but I don't thinks that's right... Feb 11, 2019 at 20:45

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.