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 be the orthocentre of a non-equilateral triangle, let be the centre of gravity, let be the centre of the circumscribed circle and let be the centre of the nine-point circle. The points then lie on a straight line (Euler's line), being the midpoint of the segment , and the pair of points harmonically subdivides the pair of points .
|||S.I. Zetel', "A new geometry of triangles" , Moscow (1962) (In Russian)|
|||D.I. Perepelkin, "A course of elementary geometry" , 1 , Moscow-Leningrad (1948) (In Russian)|
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 (giving a coordinate system) in the projective plane, cf. [a2], Sects. 18.104.22.168, 16.7.5.
|[a1]||H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)|
|[a2]||M. Berger, "Geometry" , 1–2 , Springer (1987) pp. Sects. 10.11.3, 17.5.4 (Translated from French)|
|[a3]||O. Veblen, J.W. Young, "Projective geometry" , II , Blaisdell (1946) pp. 169; 233|
Nine-point circle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nine-point_circle&oldid=18114