In this thread Jose discovered that the length between the vertex of any non equilateral triangle and the associated vertex of an equilateral triangle constructed on the same base opposite the vertex, is the same length, for all the vertices of the triangle. For lack of a better name let's call this length the Jose length of the triangle.
While playing around with this curiosity in Geogebra I discovered this length appears to be always equal to any side length of the inner Napolean triangle multiplied by $ \sqrt{3} \ $. Can anyone show why that is always the case and why the factor of $\sqrt{3} \ $? A Geogebra construction demonstrating this is here. The red segments are the Jose lengths and the green triangle is the inner Napolean triangle.
Edit: As an aside, there is another interesting relationship between the Jose lengths and inner Napolean triangle. If perpendicular bisectors are constructed on the Jose segments, a new triangle is defined, which is congruent to the the inner Napolean triangle, but "orbits" around the outside of the triangle. The Geogebra construction is here. I think this construction basically demonstrates that the Jose segments are always perpendicular to a corresponding side of the inner Napolean triangle. Its hard to tell if it is effectively a rotation or a reflection of the inner Napolean triangle or both. As the parent triangle more closely approaches being an equilateral triangle the orbital triangle more closely approaches being a point on the circumcircle of the parent triangle and when the parent triangle is exactly an equilateral triangle the location of the point is not precisely defined and appears almost random.