Nine-Point Center, Nine-Point Circle, Euler Line.

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The Nine-point center is the center of the nine-point circle. The Nine-Point Circle of triangle ABC with orthocenter H is the circle that  passes through the feet of the altitudes H_A, H_B and H_C to the three sides, the midpoints M_A, M_B and M_C of those sides, and the Euler Points E_A, E_B and E_C, which are the midpoints of the segments AH, BH, and CH, respectively. Euler line is the line passing through the orthocenter H, the nine-point center N, the centroid G, and the circumcenter O of any triangle ABC. Click the Next button above to start the step-by-step illustration.

• The nine-point center N is the midpoint of the line HO.

• The distance from the orthocenter H to the centroid G is twice the distance from the circumcenter O to the centroid G.

• The nine-point center N is the circumcenter of the medial triangle M_AM_BM_C.

• The nine-point center N is the circumcenter of the orthic triangle H_AH_BH_C.

The nine-point circle is also known as Euler's circle and Feuerbach's circle. Leonhard Euler showed in 1765 that the nine-point circle bisects any line from the orthocenter to a point on the circumcircle. In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle.