Does every set of $n$ points in the Euclidean plane contain three points $A,B,C$ such that the two segments obtained by joining $A,B$, respectively $A,C$ form an angle at least equal to $(1-2/n)\pi$ at the point $A$? (Equality is of course achieved by the vertex set of a regular $n-$gone.)

Pietro Majer's example below can be generalized and shows that $(1-2/n)\pi$ has to be replaced by a somewhat smaller constant (at least for values of $n$ which are large enough). For his example we have to take $(1-2/6)\pi$ instead of $(1-2/7)\pi$. Is the best possible constant $(1-a(n))\pi$ asymptotically substantially better, ie. can $na(n)$ become for example arbitrarily large for $n$ large enough? (It is of course obvious that $a(n)$ is decreasing but how fast?)

Update: For $n=5$, one can get arbitrarily close to $(1-1/4)\pi$: Take a right-angled isocele triangle. Split the right-angled-vertex infinitesimally along a line parallel to the longest side of the initial triangle and add an additional point on the symmetry axis very high above the two infinitesimal points.