page_title: Nine-point circle
section_title: Other properties of the nine-point circle
hierarchical_section_title: Nine-point circle / Other properties of the nine-point circle
caption_attribution_description: English: ABCD is a cyclic quadrilateral. EFG is the diagonal triangle of ABCD. The point T of intersection of the bimedians of ABCD belongs to the nine-point circle of EFG.
context_page_description: In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
  The midpoint of each side of the triangle
  The foot of each altitude
  The midpoint of the line segment from each vertex of the triangle to the orthocenter.
  The nine-point circle is also known as Feuerbach's circle, Euler's circle, Terquem's circle, the six-points circle, the twelve-points circle, the n-point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle.
context_section_description: The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle.
  Figure 3
  A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.
  Figure 4
  The center N of the nine-point circle bisects a segment from the orthocenter H to the circumcenter O (making the orthocenter a center of dilation to both circles):
  ON = NH.
  The nine-point center N is one-fourth of the way along the Euler line from the centroid G to the orthocenter H:
  HN = 3NG.
  Let  be the nine-point circle of the diagonal triangle of a cyclic quadrilateral. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle.
  The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's medial triangle (with vertices at the midpoints of the sides of the reference triangle) and its orthic triangle (with vertices at the feet of the reference triangle's altitudes).
  The center of all rectangular hyperbolas that pass through the vertices of a triangle lies on its nine-point circle. Examples include the well-known rectangular hyperbolas of Keipert, Jeřábek and Feuerbach. This fact is known as the Feuerbach conic theorem.
  If an orthocentric system of four points A, B, C and H is given, then the four triangles formed by any combination of three distinct points of that system all share the same nine-point circle. This is a consequence of symmetry: the sides of one triangle adjacent to a vertex that is an orthocenter to another triangle are segments from that second triangle. A third midpoint lies on their common side. (The same 'midpoints' defining separate nine-point circles, those circles must be concurrent.)
  Consequently, these four triangles have circumcircles with identical radii. Let N represent the common nine-point center and P be an arbitrary point in the plane of the orthocentric system. Then
  NA²+NB²+NC²+NH² = 3R²
  where R is the common circumradius; and if
  PA²+PB²+PC²+PH² = K²,
  where K is kept constant, then the locus of P is a circle centered at N with a radius . As P approaches N the locus of P for the corresponding constant K, collapses onto N the nine-point center. Furthermore the nine-point circle is the locus of P such that
  PA²+PB²+PC²+PH² = 4R².
  The centers of the incircle and excircles of a triangle form an orthocentric system. The nine-point circle created for that orthocentric system is the circumcircle of the original triangle. The feet of the altitudes in the orthocentric system are the vertices of the original triangle.
  If four arbitrary points A, B, C, D are given that do not form an orthocentric system, then the nine-point circles of ABC, BCD, CDA and DAB concur at a point. The remaining six intersection points of these nine-point circles each concur with the midpoints of the four triangles. Remarkably, there exists a unique nine-point conic, centered at the centroid of these four arbitrary points, that passes through all seven points of intersection of these nine-point circles. Furthermore because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular circumconic, centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles.
  If four points A, B, C, D are given that form a cyclic quadrilateral, then the nine-point circles of ABC, BCD, CDA and DAB concur at the anticenter of the cyclic quadrilateral. The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The nine-point circles form a set of four Johnson circles. Consequently the four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral. Furthermore the cyclic quadrilateral formed from the four nine-pont centers is homothetic to the reference cyclic quadrilateral ABCD by a factor of  −¹/₂   and its homothetic center (N) lies on the 
caption: ABCD is a cyclic quadrilateral. EFG is the diagonal triangle of ABCD. The point T of intersection of the bimedians of ABCD belongs to the nine-point circle of EFG.
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