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The nine-point center is indexed as X ( 5 ), the Feuerbach point, as X ( 11 ), the center of the Kiepert hyperbola as X ( 115 ), and the center of the Jeřábek hyperbola as X ( 125 ).
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nine-point and center
The orthocenter, along with the centroid, circumcenter and center of the nine-point circle all lie on a single line, known as the Euler line.
The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
* 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.
Let N represent the common nine-point center and P be an arbitrary point in the plane of the orthocentric system.
As P approaches N the locus of P for the corresponding constant K, collapses onto N the nine-point center.
* Trilinear coordinates for the nine-point center are cos ( B − C ): cos ( C − A ): cos ( A − B )
Where O, O < sub > 4 </ sub > and A < sub > 4 </ sub > are the nine-point center, circumcenter orthocenter, respectively of the triangle formed from other three orthocentric points A < sub > 1 </ sub >, A < sub > 2 </ sub > and A < sub > 3 </ sub >.
The center of this common nine-point circle lies at the centroid of the four orthocentric points.
The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes.
This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points.
The orthocentric point that becomes the incenter of the orthic triangle is that orthocentric point closest to the common nine-point center.
Let vectors < u > a </ u >, < u > b </ u >, < u > c </ u > and < u > h </ u > determine the position of each of the four orthocentric points and let < u > n </ u > = (< u > a </ u > + < u > b </ u > + < u > c </ u > + < u > h </ u >) / 4 be the position vector of N, the common nine-point center.
Join each of the four orthocentric points to their common nine-point center and extend them into four lines.
This generated othocentric system is always homothetic to the original system of four points with the common nine-point center as the homothetic center and α the ratio of similitude.
nine-point and is
In geometry, the nine-point circle is a circle that can be constructed for any given triangle.
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.
Although he is credited for its discovery, Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six point circle, recognizing the significance of the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle.
The nine-point circle is tangent to the incircle and excircle s
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle ; this result is known as Feuerbach's theorem.
The point at which the incircle and the nine-point circle touch is often referred to as the Feuerbach point.
* The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle.
* 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.
Furthermore the nine-point circle is the locus of P such that PA < sup > 2 </ sup >+ PB < sup > 2 </ sup >+ PC < sup > 2 </ sup >+ PH < sup > 2 </ sup > = 4R < sup > 2 </ sup >.
The nine-point circle created for that orthocentric system is the circumcircle of the original triangle.
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.
* Letting x: y: z be a variable point in trilinear coordinates, an equation for the nine-point circle is
In 1822 he wrote a small book on mathematics noted mainly for a theorem on the nine-point circle, which is now known as Feuerbach's theorem.
The circle tangent to all three of the excircles as well as the incircle is known as the nine-point circle.
The point where the nine-point circle touches the incircle is known as the Feuerbach point.
When P is chosen as the circumcenter O, then α = − 1 and the generated orthocentric system is congruent to the original system as well as being a reflection of it about the nine-point center.
nine-point and X
* The points X ( 13 ), X ( 14 ), circumcenter, nine-point center lie on a Lester circle.
nine-point and ),
Euler's line ( red ) is a straight line through the centroid ( orange ), orthocenter ( blue ), circumcenter ( green ) and center of the nine-point circle ( red ).
nine-point and Feuerbach
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.
* Karl Feuerbach describes the nine-point circle of a triangle.
nine-point and point
* A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.
* 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.
Note also that the one point on the nine-point circle that is the center of this rectangular hyperbola will have four different definitions dependent on which of the four possible triangles is used as the reference triangle.
These four orthic inconics also share the same Brianchon point, H, the orthocentric point closest to the common nine-point center.
It passes through the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
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