# Why three $60^{\circ}$ corner angles cannot be drawn to make a triangle?

On a spherical surface $K=+1$ a curvilinear triangle with $60^{\circ}$ at each vertex cannot be drawn because the spherical excess $\pi - 3 A =0$ and so its area vanishes. (By virtue of Gauss-Bonnet theorem).

Similarly on a pseudospherical surface $K=-1$ a curvilinear triangle with $60^{\circ}$ at each vertex cannot be drawn because the pseudospherical defect $3 A - \pi =0$ and vanish.

Such a construction is impossible however small the size of geodesic triangle be. It requires that the lines should be concurrent when applied to elliptic/hyperbolic situations like a star made by three concurrent geodesic arcs.

What actually prevents formation of such "triangles" of vanishing enclosed area ?

EDIT1:

An attempt to go further in such a search: By drawing three equal geodesic arcs across cuspidal equator $(K=-1)$ hypo pseudosphere roughly as per sketch we can/may have a zero area enclosed between them surrounding singularity of cone vertex,of course with additional and appropriate interpretation/condition. Corner angles at cuspidal equator is $\pi/3.$

Two related real constants $(b,m)$ arise in geodesic calculation which should be found from a definite integral.

$$\int_{b}^{m} \frac{dr}{r\sqrt{((r/b)^2-1)(1-m^2+r^2)}} = \frac{\pi}{3}$$

Request your help in their determination.

If you have a line and on that line make two $60^\circ\!$ angles pointed towards eachother (the way you would make an equilateral triangle in a flat plane), the two legs will curve away from one another and meet at an angle smaller than $60^\circ\!\!$, if they meet at all. This is because $K<0$.
Note that you have the same phenomenon on the sphere $S^2$. I'd say that the case $\kappa=0$ of the Euclidean plane is special, in so far as the angle sum of triangles (resp., the total geodesic curvature along a smooth boundary) is $\pi$ (resp., $\pm2\pi$), independent of the triangle or shape in question. This has to do with the fact that in the Euclidean plane we have the group of scalings at our disposal. A scaling changes the areas, but not the total geodesic curvature.