This exploration focuses
on the center (**N**) of the Nine-Point Circle of
any triangle and its relationship to the centroid, orthocenter,
circumcenter, and incenter for different shaped triangles.

The Nine-Point Circle is a circle that passes through the three midpoints of the sides of a triangle, the three feet of the altitudes of the triangle, and the three mid-points of the segments from the respective vertices to the orthocenter.

The center of the Nine-Point
Circle is labeled as **N**,
the centroid is **G**, the orthocenter is **H**, the circumcenter is **C**, and the incenter is **I** for the different shaped triangles
of triangle **RST**.

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**N**) of the Nine-Point Circle

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**H**,
**I**, **G**,
and **C**) are collinear with **N**

**N**
is the midpoint of segment **SC** (the altitude of the right
angle) and of segment **HC** (the segment of the orthocenter
to circumcenter)

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**N**
is the midpoint of segment **HC**

**N**,
**G**, **C**,
and **H** are still collinear (**I** is no longer collinear)

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**N** is the midpoint of segment

**N**,

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**N** is the midpoint of segment

**N**,

**N** will fall to the exterior of
the triangle.

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**N** is the midpoint of segment

**N**,

**N**
is concurrent with the vertex of the obtuse angle

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