A Euler line is a line that goes straight through all three of the following: the orthocenter, the centroid, and the circumcenter. If one of the triangle's points is moved, the whole triangle will be different, making the orthocenter, the centroid, and the circumcenter change in correspondence to the change of the triangle.
Understanding the Euler line: Change C_1, which is currently C_1(-5, -8), to (-1.19, -5.67). Now click the Hide/Show Euler line check box to show the Euler line. Observe the different position.