Nine-point Circle - Other Properties of The Nine-point Circle

Other Properties of The Nine-point Circle

• 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 of any nine-point circle (the nine-point center) lies on the corresponding triangle's Euler line, at the midpoint between that triangle's orthocenter and circumcenter.

• The nine-point center lies at the centroid of four points comprising the triangle's three vertices and its orthocenter.

• Of the nine points, the three midpoints of line segments between the vertices and the orthocenter are reflections of the triangle's midpoints about its nine-point center.

• 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 cylic 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 line connecting the circumcenter (O) to the anticenter (M) where ON = 2NM.

• The orthopole of lines passing through the circumcenter lie on the nine-point circle.

• Trilinear coordinates for the nine-point center are cos (B − C) : cos (C − A) : cos (A − B)

• Trilinear coordinates for the Feuerbach point are 1 − cos (B − C) : 1 − cos (C − A) : 1 − cos (A − B)

• Trilinear coordinates for the center of the Kiepert hyperbola are (b2 − c2)2/a : (c2 − a2)2/b : (a2 − b2)2/c

• Trilinear coordinates for the center of the Jeřábek hyperbola are cos A sin2(B − C) : cos B sin2(C − A) : cos C sin2(A − B)

• Letting x : y : z be a variable point in trilinear coordinates, an equation for the nine-point circle is

x2sin 2A + y2sin 2B + z2sin 2C − 2(yz sin A + zx sin B + xy sin C) = 0.