# Nine-point circle

2020 Mathematics Subject Classification: Primary: 51M15 [MSN][ZBL]

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\$.

Figure: n066750a

#### References

 [1] S.I. Zetel', "A new geometry of triangles" , Moscow (1962) (In Russian) [2] D.I. Perepelkin, "A course of elementary geometry" , 1 , Moscow-Leningrad (1948) (In Russian)