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.

  • $\begingroup$ 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. $\endgroup$ Feb 11, 2019 at 19:05
  • $\begingroup$ The argument here showing how to construct the tangents may offer a clue to a proof. cut-the-knot.org/Curriculum/Geometry/GeoGebra/… $\endgroup$ Feb 11, 2019 at 19:23
  • $\begingroup$ @Mark Bennet He wants a maximal number of tangents. $\endgroup$ Feb 11, 2019 at 19:25
  • $\begingroup$ @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. $\endgroup$
    – Narasimham
    Feb 11, 2019 at 19:55
  • $\begingroup$ @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... $\endgroup$ Feb 11, 2019 at 20:45

1 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.

  • $\begingroup$ Maybe I didn't phrase it the best way, but what I meant with not using Calculus was actually not using a coordenate system and equations for the circles. Do you know of some other way? $\endgroup$ Feb 11, 2019 at 19:16
  • $\begingroup$ @J. Dionisio It's obvious geometrically. Just draw two circles. $\endgroup$ Feb 11, 2019 at 19:20
  • $\begingroup$ @J. Dionisio Actually, I did not use equations of circles. $\endgroup$ Feb 11, 2019 at 19:28
  • $\begingroup$ Yes you did not, I apologize. And I know it's obvious, but some proof must exist, no? $\endgroup$ Feb 11, 2019 at 19:48
  • $\begingroup$ I wrote this proof. This system gives four tangents maximum. $\endgroup$ Feb 11, 2019 at 19:52

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