Nine Point Circle
10. The Nine-Point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to orthocenter. Construct the nine points, locate the center (N) and construct the nine point circle.
First we construct a triangle ABC. Then find the midpoints of each side and the orthocenter of the triangle (where the altitudes of the vertices meet).
Now we label the points where the feet of the altitudes meet on each side of the triangle. Since the above triangle is obtuse only one altitude meets a side of the triangle. Now we label the midpoints of the segments that are the vertices to the point of intersection of the altitudes. Then connect the medians of the triangle.
Now we draw perpendicular at the midpoints of the sides of the medial triangle (triangle by connecting the medians). Then find the intersection of these perpendicular lines. This is the center N of the nine point triangle. Our nine point circle has a radius of N to any of the sidesŐ medians.
Drawing out the circle we see our nine points that are on the circle.
Three of the points are the medians of the sides, three are the points where the altitudes meet the sides and three the midpoints between the vertices of the triangles and the orthocenter.
Let us now see how the nine point circle behaves with different types of triangles.