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.

**(Return to Write-Up
8)**

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

Why are some of the points the same?

A Right Triangle

An Obtuse Triangle

Click **here** to view the Nine-Point
circle of varying triangles.

Click **here** to see how the
center of the Nine-Point circle (N) compares to the other centers**
**{circumcenter (**C**), centroid (**G**), orthocenter
(**H**), incenter (**I**)}** **of varying traingles.

**Return to Philippa's EMT Homepage
**