SCHWEITZER 1954 Problem 10

Given a triangle $ABC$ , construct outwards over the sides $AB, BC, CA$ similiar isosceles triangles $ABC_{1}, BCA_{1}$ and $CAB_{1}$ . Prove that the straight lines $AA_{1}. BB_{1}$ and $CC_{1}$ are concurrent. Is this statemente true in elliptic and hyperbolic geometry, too? (G. 19)