TITLE: Is there a map from a segment to a triangle?
QUESTION [6 upvotes]: Is there a conformal map from a sector in the circle with 60 degrees angle to a equilateral  triange that maps the points 0, 1 and $exp (\frac{\pi i}{3})$ to the vertices of a triangle?

REPLY [3 votes]: I realized that my question is quiet easy, we can just map a segment to a circle as well as a triangle to a circle by Riman theorem. Then the property we do need that the vertices  of triangle map to the same points as "vertices" of a segment. That can be done by adding an automorphism of a circle, that maps three first images on the boundary to the three second images on the boundary. Then we just have to take a composition.
So, the form of the domains in this problem is not important as far as it concerns only three points on the boundary.