In the previous post, we have seen that the orthocenter, circumcenter, and centroid of a triangle lie on the same line. In this post, we celebrate another property of triangles related to the previous post.
Consider the following points.
(a) The intersections of the altitudes and the base.
(b) The intersections of medians and the base. A median is a segment that connecting the vertex and the midpoint of the opposite base.
(c) The midpoints of the orthocenter (the intersection of the altitudes) and the the three vertices.
The nine points in (a), (b), and (c) lie on a circle.
The nine-point circle is also called the Euler’s circle or Feuerbach’s circle.