Assignment 4

**Constructing the Nine-Point Circle and its Related Triangles**

by Margaret Trandel

**Step 1:** Construct a triangle ABC

**Step 2:** Construct the three altitudes of an triangle ABC.

[Remember: an altitude of a triangle is a line segmet from the vertex of the
triangle to the perpendicular foot of the opposite side (here A', B', and
C').]

The three altitudes concur at the orthocenter. Let's refer to the
orthocenter as "H".

**Step 3:** Connect points A', B', and C' to make another triangle, which is
the orthic triangle of triangle ABC.

**Step 4:** Find the midpoints of the sides of the original triangle
A", B", and C" and connect them to make another triangle, which is the medial
triangle of ABC.

**Step 5:** Construct the midpoints from the orthocenter to each
vertex and label them M1, M2, and M3.

**Step 6:** Connect these three points, M1, M2, and M3, to form a
third triangle -- which is called the image triangle (or the orthocenter
midsegment triangle) of ABC.

**Step 7:** Now we can connect nine points -- M1, M2, M3, A', B',
C', A", B", and C" -- to form the nine-point circle.

**Note:** The center of the nine-point circle, C9, is the midpoint
between the orthocenter H and the incenter I.

[Remember, the incenter is the point of concurrency of the perpendicular
bisectors of each side of the triangle.]

**Key:**

**H** = Orthocenter

**I** = Incenter

**C9** = Center of Nine-Point Circle

**A**, **B**, and **C** are vertices of the original triangle

**A'**, **B'**, and **C'** are the triangles altitudes

**A"**, **B"**, and **C"** are the midpoints of the triangle sides

**M1**, **M2**, and **M3** are the midpoints between the orthocenters and the
three vertices of the triangle.