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There are 3 tangent circles:

enter image description here

Now we move circles A and B such that they intersect C but do not intersect each other:

enter image description here

In the figure, the angle near C increases. Will this always be the case?

I tried to prove this using the law of cosines:

$$\cos{\angle{C}} = \frac{a^2+b^2-c^2}{2ab}$$

When the circles are tangent, the following equations hold (where $r_X$ is the radius of circle $X$):

$$a = R_B+R_C\ \ \ \ \ b=R_C+R_A\ \ \ \ c=R_A+R_B$$

But after we move them:

$$a' < R_B+R_C\ \ \ \ \ b'<R_C+R_A\ \ \ \ c'>R_A+R_B$$

$a$ and $b$ decrease, while $c$ increases. The nominator of $\cos{\angle{C}}$ indeed decreases, but, so does the denominator, so this approach doesn't lead to a proof...

Is it true that $\cos{\angle{C}}$ always increases, or maybe in some cases it can decrease?

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you can use geogebra and play with the 3 circles. Just click on the link below: geogebratube.org/student/m82249 –  ulead86 Feb 3 at 13:35
    
@ulead86 Thank you! This site looks very useful. –  Erel Segal Halevi Feb 4 at 10:48

1 Answer 1

up vote 3 down vote accepted

Is it true that $\cos\angle C$ always increases, or maybe in some cases it can decrease?

It can decrease.

Instead of moving $A$ and $B$, you can also move $C$. At least those cases where the circles around $A$ and $B$ still touch are covered by this as well. In order to estimate the change in angle, draw the circumcircle around your old triangle $\triangle ABC$. If you manage to move $C$ to a new position $C'$ outside that circle, then its angle to $A$ and $B$ will decrease. This is a consequence of the inscribed angle theorem. The following example shows that you can really do so, while still intersecting both the other circles.

Counterexample

Experiments suggest that you need $\angle C$ to be pretty small for this to work, though. So an interesting follow-up question (probably best posted separately but with links to this question here) would ask for the largest $\angle C$ such that this still works.

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Thank you! Indeed, by playing with ulead84's geogebra page: geogebratube.org/student/m82249 I could make $\angle{C}$ approach 0 by moving C below the center of the lower circle such that the three centers are almost colinear, and could also make it approach 180 degrees by moving C between the centers such that the three centers are almost colinear. So, it seems there is not really a good bound on $\angle{C}$. –  Erel Segal Halevi Feb 4 at 11:03

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