**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*NA*2+*NB*2+*NC*2+*NH*2 =*3R*2 where*R*is the common circumradius and if*PA*2+*PB*2+*PC*2+*PH*2 =*K*2, 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*PA*2+*PB*2+*PC*2+*PH*2 =*4R*2.

- 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/*and its homothetic center_{2}*(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 (
*b*2*− c*2)2/*a*: (*c*2 −*a*2)2/*b*: (*a*2 −*b*2)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

*x*2sin*2A*+*y*2sin 2*B*+*z*2sin 2*C*− 2(*y*z sin*A*+*zx*sin*B*+*xy*sin*C*) = 0.

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### Famous quotes containing the words circle and/or properties:

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—Ralph Waldo Emerson (1803–1882)

“A drop of water has the *properties* of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”

—Ralph Waldo Emerson (1803–1882)

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