Complete three-ray toric surfaces with at most one singular point ($\mathbb{P}(1,1,r),r\ge 1$)

Two neighboring smooth cones can be normalized so that the three rays are $(1,0),(0,1),(-1,-r)$, with $r\ge 1$. Completeness forces the negative sign and their primitive relation has weights $(1,r,1)$. Thus these surfaces are precisely the weighted projective planes $\mathbb{P}(1,1,r)$. For $r=1$ the surface is the projective plane; for $r>1$ its unique singularity is the cyclic quotient surface singularity $\frac{1}{r}(1,1)$.