Draw a triangle, any triangle (although it may be best to start with an acute triangle).
1) Mark the midpoints of each side (3 points). See Figure 1.
2) Drop an altitude from each vertex to the opposite side, and mark the points where the altitudes intersect the opposite side. (If the triangle is obtuse, an altitude will be outside the triangle, so extend the opposite side until it intersects.) See Figure 2.
3) Notice that the altitudes intersect at a common point. Mark the midpoint between each vertex and this common point. See Figure 3.
No matter what triangle you start with, these nine points all lie on a perfect circle!
Even simple geometry still has some surprises in store! This result was known by Euler in 1765, but rediscovered by Feuerbach in 1822.