The max arc length for 3 symmetrical circles to intersect Consider the following:

3 Identical circles, intersecting in a way so that all 3 arc lengths outside are the same length.
Now here is a similar situation:

The arc length for the circles got bigger, but it is still equal on all side and symmetrical, also the circles are the same size.
Again, L1=L2=L3, R1=R2=R3.
Is there a way to keep going like this, until it is no longer possible to have a symmetrical situation like this. What would the max arc length be, in terms of the radius ? Hope I was clear!
Thanks all
P.S: Excuse my drawings, It may not be exactly as I described, I would gladly accept any software idea to better draw geometrical figures.
 A: When you write “symmetrical”, what you mean is mirror symmetry, right?
Let's for the moment assume all radii are $R=1$ and the centers of the circles are located at $(a,0)$, $(-a,0)$ and $(0,b)$ for some $a,b>0$. That ensures the symmetry, so now you have to formulate the condition for equal arc length.
I did that using some lengthy computation: formulated points with variables as conditions, then formulated all on-circle conditions as polynomial equations. I also formulated the equal-arclength condition as a polynomial equation, and I did so once using the determinant (which scales with the sine of the angle) and once using the dot product (which scales with the cosine). I used variable elimination techniques to obtain a relation between $a$ and $b$, and factorized that. Lastly I looked for factors common to both equal-angle formulations, and among those chose the only one which makes sense for intersecting circles, which turns out to be $3a^2=b^2$.
Perhaps there are easier ways to obtain this. After all, all that this condition tells you is the fact that the centers of the circles have to be arranged such that they form a isosceles triangle. Using this condition, you can move freely between the “all circles equal” and the “two circles just touch in one point” extremes:

I did the above animation with sage, but for interactive geometry construction I'd suggest Cinderella.
The maximal arc length can be obtained from the merely-touching situation. In that case, the circle centers form an equilateral triangle and the touching points are on the edges of that triangle, with $\frac13\pi$ interior angle and $\frac53\pi$ exterior angle. So the maximal arc length is
$$L=\frac53\pi R$$
