Nine Point Circle

Rozina Essani

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.

Isosceles Triangle:

Right Triangle:

Obtuse triangle