Situation 1a:  The triangle formed by joining the midpoints of the sides of an existing triangle is called a medial triangle.  Is the circumcenter of the existing triangle also the orthocenter of the medial triangle?

Situation 1b:  Is the centroid of the existing triangle also the centroid of the medial triangle?

Situation 1c:  Are the circumcenter of the existing triangle, the circumcenter of the medial triangle and the centroids of both triangles collinear?

Situation :  Locating the nine-point center of a triangle.

1.  Draw any triangle and label the vertices A, B and C.
    (Hint: start with a relatively large acute triangle.)

2.  Locate by construction the midpoints of the sides of the triangle.  Label the midpoints M1, M2, and M3.

3.  Locate by construction the altitudes to the sides of the triangles.  Label the base of each altitude H1, H2, and H3.

4.  Label the orthocenter, P.

5.  Locate by construction the midpoints of segments , and .  Label these midpoints R, S and T respectively.

6.  Construct a circle which passes through all nine of the new points you have just located (points M1, M2, M3, H1, H2, H3, R, S, T ).  Label the center of the circle O.

This circle is called the nine-point circle of .  The center of the circle is called the nine-point center of the triangle.  The nine-point circle is also known as Euler's circle.