Leonhard Euler in 1765 showed that for any triangle, the centroid, circumcenter, and orthocenter are collinear. In fact, the line that passes through these triangle centers also passes through several other notable points such as the incenter, the nine-point center, the de Longchamps point, and others. And, remarkably, when you change the shape of the triangle, the relative distances between the centers is unchanged.

Construct an interface to display a triangle with dynamic vertices together with the triangles centroid, circumcenter, orthocenter, and the Euler line. Give distinct colors to each of the four sets: incenter and angle bisectors, medians and centroid, perpendicular bisectors and circumcenter and circumcircle, altitudes and orthocenter. Add a legend that identifies the objects by color.