View source for Nine-point circle

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''Euler circle, Feuerbach circle''

A circle whose periphery contains the midpoints of the sides of a triangle, the bases of its altitudes, and the midpoints of the segment connecting the orthocentre of the triangle with the vertices. Its radius is equal to one-half of the radius of the circle circumscribed about the triangle. The nine-point circle of a triangle is tangent to the circle inscribed in it and to the three escribed circles. Let $H$ be the orthocentre of a non-equilateral triangle, let $T$ be the centre of gravity, let $O$ be the centre of the circumscribed circle and let $E$ be the centre of the nine-point circle. The points $H,T,O,E$ then lie on a straight line (the [[Euler line]]), $E$ being the midpoint of the segment $HO$, and the pair of points $H,T$ harmonically subdivides the pair of points $O,E$.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066750a.gif" />

Figure: n066750a

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S.I. Zetel',   "A new geometry of triangles" , Moscow  (1962)  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  D.I. Perepelkin,   "A course of elementary geometry" , '''1''' , Moscow-Leningrad  (1948)  (In Russian)</TD></TR>
</table>



====Comments====
Sometimes the nine-point circle is referred to as the Feuerbach circle. The fact that the nine-point circle is tangent to the inscribed circle and the three escribed circles is Feuerbach's theorem.

More generally one has the nine-point conic and the eleven-point conic determined by a projective base $\{a,b,c,d\}$ (giving a coordinate system) in the projective plane, cf. [[#References|[a2]]], Sects. 16.5.5.1, 16.7.5.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1961)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''1–2''' , Springer  (1987)  pp. Sects. 10.11.3, 17.5.4  (Translated from French)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Veblen,   J.W. Young,   "Projective geometry" , '''II''' , Blaisdell  (1946)  pp. 169; 233</TD></TR>
</table>