To construct the nine point circle we begin with any triangle.

The first three points on the circle will be the midpoints of the sides of the triangle.

The next three points on the circle will be the feet of the altitudes of the triangle.

For the final set of three points, we need to find the orthocenter of the triangle. The orthocenter is the intersection of the three perpendicular segments from one vertex to the opposite line segment. These perpendicular segments will be the altitudes of the triangle when they are inside the triangle.

The final three points will be the midpoints of the segments from the vertices to the orthocenter.

Now we have all nine points of our nine point circle!

The next step is to construct the circle. Another characteristic of the circle is that the center of the nine point circle is the midpoint of the segment connecting the orthocenter and the circumcenter.

The circumcenter is a point equidistant from each of the three vertices. The circumcenter of a triangle is constructed from the point of intersection of each side of the triangle's perpendicular bisector. The circumcenter of a triangle is labeled with a C.

The center of the nine point circle:

Now, we can construct a circle using our center and any of the nine points.

Notice how the circle goes throught each of the nine points.

To see the nine point circles with other triangles, use the tool on the GSP file.

Below are images of some different triangle/nine point circles you may see.

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