Philippa M. Rhodes

## Write-up 4

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 the orthocenter. Construct the nine points, locate the center (N) and construct the Nine-Point circle.

First, draw triangle ABC and locate the mid-point of each side.

The orthocenter (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side.
By constructing the orthocenter of triangle ABC,

we are able to locate the next three points, the feet of the altitudes (the point of intersection of altitude and the opposite side). In order to construct the last three points, simply locate the mid-points of the segments from the respective vertices to the orthocenter.

The circumcenter of a triangle is the point in the plane equidistant from the three vertices of the triangle. Also, it is the center of the circumcircle of the triangle. Since we want the Nine-Point circle to pass through all nine points (including R, S, and T ), we can construct the circumcircle of triangle RST.

We notice that the circumcenter of triangle RST is the center of the Nine-Point circle of triangle ABC. So, we label it N.

Now let's look at the Nine-Point circle of other triangles.

An Equilateral Triangle

Why are some of the points the same?

A Right Triangle

An Obtuse Triangle