9 Point Circle

(Centroid, Orthocenter, Circumcenter, and Incenter)

This exploration focuses on the center (N) of the Nine-Point Circle of any triangle and its relationship to the centroid, orthocenter, circumcenter, and incenter for different shaped triangles.

The Nine-Point Circle is a circle that passes through the three midpoints of the sides of a triangle, the three feet of the altitudes of the triangle, and the three mid-points of the segments from the respective vertices to the orthocenter.

The center of the Nine-Point Circle is labeled as N, the centroid is G, the orthocenter is H, the circumcenter is C, and the incenter is I for the different shaped triangles of triangle RST.

Equilateral

all points are concurrent with the center (N) of the Nine-Point Circle

Right-Isosceles

all points (H, I, G, and C) are collinear with N

notice that N is the midpoint of segment SC (the altitude of the right angle) and of segment HC (the segment of the orthocenter to circumcenter)

Right-Scalene Triangle

N is the midpoint of segment HC

N, G, C, and H are still collinear (I is no longer collinear)

Acute-Scalene Triangle

N is the midpoint of segment HC

N, G, C, and H are still collinear

Obtuse-Scalene Triangle

N is the midpoint of segment HC

N, G, C, and H are still collinear

As the obtuse angle gets larger, N will fall to the exterior of the triangle.

Obtuse-Isosceles Triangle