Nine-Point Center

The center $F$ (or $N$) of the Nine-Point Circle. It has Triangle Center Function

$$\alpha = \cos(B-C)=\cos A+2\cos B\cos C$$

$$= bc[a^2b^2+a^2c^2+(b^2-c^2)^2],$$

and is the Midpoint of the line between the Circumcenter $C$ and Orthocenter $H$. It lies on the Euler Line.

See also Euler Line, Nine-Point Circle, Nine-Point Conic

References

Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 624, 1970.

Dixon, R. Mathographics. New York: Dover, pp. 57-58, 1991.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Kimberling, C. ``Nine-Point Center.'' http://cedar.evansville.edu/~ck6/tcenters/class/npcenter.html.