Let $P$ be a point where three arcs of circle meet at equal angles (120 degrees). Suppose that the sum of the curvatures (with sign given by orientation) of the three arcs is zero.
Is it true that this property is preserved under a moebius transformation? Angles are preserved, but what about the sum of curvatures?
What if we don't know anything about the angles? Is the sum of curvatures anyway preserved (if it is zero)?
What seems to be true is that given three arcs coming out from a point $O$ with equal angles (120 degrees) and with curvatures whose sum is zero, then the arcs meet (all three) at another point $P$. This can be found in Lemma 8.1 of Cox, Harrison, Hutchings "The shortest enclosure of three connected areas in $\mathbb R^2$" (Real Anal. Exch.)". The proof is by direct computation, left to the reader.
It seems natural that also the converse is true i.e.: if the arcs all meet in another single point, then the sum of curvatures is zero. This would solve the problem, since moebius transform preserve this condition.
It seems to me that these properties should be known, because they seem quite important... don't they?