Take any triangle (as shown here). It has 3 kinds of "centers":
1. You can look at its usual center of mass. (Where it balances out.)
2. You can look at its orthocenter -- that is, the point where the perpendicular altitutes all cross. (And all 3 altitudes do cross at the same point.)
3. You can look at the center of the circle that goes thru its corners. (There's only one such circle.)
It turns out that all these three points are on the same line (i.e., they're colinear). That line is called the Euler line of the triangle. (The one exception is when all these points are the same point when the triangle is an equilateral.) I don't think this is hard to work out and show, but this result is interesting to know.