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In this paper we introduce the interpolationdegeneration strategy to study KhlerEinstein 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] that admit a conical KhlerEinstein metric form a connected interval, and by degeneration we determine the boundary of the interval in some important cases. As a first application, we show that there exists a KhlerEinstein metric on P 2 with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in (/2, 2]. When the angle is 2/3 this proves the existence of a SasakiEinstein metric on the link of a three dimensional <i>A</i> ₂ singularity, and thus answers a question posed by GauntlettMartelliSparksYau. As a second application we prove a version of Donaldsons conjecture about conical Khler

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