This article describes an investigation into Kiepert lines, and leads to some surprising and little-known relationships between the Fermat, Napoleon and Vecten points of a triangle.If we draw similar isosceles triangles A'BC, B'CA and C'AB outwards on the sides of a given scalene triangle ABC as in Figure 1, Kiepert's theorem tells us that the lines A'A, B'B and C'C meet in a single point - a Kiepert point [1, Chapter 11]. Since its position depends on the common base angles θ of the isosceles triangles, I label it K(θ), taking θ as the parameter of this point.