Every triangle has a unique point, called the conformal center, from which a random (Brownian motion) path is equally likely to first exit the triangle through each of its three sides. We use concepts from complex analysis, including harmonic measure and the Schwarz-Christoffel map, to locate this point. We could not obtain an elementary closed form expression for the conformal center, but we show some series expressions for its coordinates. These expressions yield some new hypergeometric series identities. Using Maple in conjunction with a homemade Java program, we numerically evaluated these series expressions and compared the conformal center to the known geometric triangle centers. Although the conformal center does not exactly coincide with any of these other centers, it appears to always lie very close to the Second Morley point. We empirically quantify the distance between these points in two different ways. In addition to triangles, certain other special polygons and circles also have conformal centers. We discuss how to determine whether such a center exists, and where it will be found.