The Nine Point Circle illustrated above is constructed from and triangle and three different sets of points relating to the triangle.  An explanation of the three different components of the Nine Point Circle follow.

• The midpoints of the three sides of the triangle ABC as illustrated below with points M, N, and L.

To construct the Nine-Point Circle a triangle is formed from the three midpoints and the intersection of the perpendicular bisectors forms the center of the Nine-Point Circle.  Also, the segment from these midpoints to the center point U are a radius of the circle.  Below is a diagram of the three midpoints and the center U.

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• The feet of the altitudes of the vertices as illustrated below with points D, E, and F

The altitudes of a triangle are the perpendicular segments between the vertices and opposing sides.

• The three Euler Points as illustrated below with points X, Y, and Z

The Euler points are the midpoints of the segments AH, BH, and CH, with H being the orthocenter.
The orthocenter is the intersection of the altitudes.

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More About the Nine-Point Circle

• Feurerbach's Theorem-The Nine-Point Circle is tangent to the incircle of triangle ABC and  its three excircles (see diagram below)

• The Nine-Point Center U lies on the Euler line of triangle ABC, which also passes through the orthocenter, the circumcenter, and the centroid (see diagram below).
• The segment from the orthocenter H to the Nine-Point Circle center U is equal to one half the length of the segment from H to the circumcenter O, or:

HU=1/2(HO)
See below for some illustrations

• The Nine-Point Circle Theorem-The radius of the Nine-Point Circle is one-half the circumradius of triangle ABC.  In addition, the cross ratio (HG,UO)=1 or

(HU x GO)/(HO x GU)

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