  In this paper we introduce the "interpolation-degneration" strategy to study
Kahler-Einstein metrics on a smooth Fano manifold with cone singularities along
a smooth divisor that is proportional to the anti-canonical divisor. By
"interpolation" we show the angles in $(0, 2\pi]$ that admit a conical
Kahler-Einstein metric form an interval; and by "degeneration" we figure out
the boundary of the interval. As a first application, we show that there exists
a Kahler-Einstein metric on $P^2$ with cone singularity along a smooth conic
(degree 2) curve if and only if the angle is in $(\pi/2, 2\pi]$. When the angle
is $2\pi/3$ this proves the existence of a Sasaki-Einstein metric on the link
of a three dimensional $A_2$ singularity, and thus answers a problem posed by
Gauntlett-Martelli-Sparks-Yau. As a second application we prove a version of
Donaldson's conjecture about conical Kahler-Einstein metrics in the toric case
using Song-Wang's recent existence result of toric invariant conical
Kahler-Einstein metrics.
