In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. The nine-point center is listed as point X(5) in Clark Kimberling's Encyclopedia of Triangle Centers.[1][2]
A triangle showing its circumcircle and circumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue)
The nine-point center N lies on the Euler line of its triangle, at the midpoint between that triangle's orthocenterH and circumcenterO. The centroidG also lies on the same line, 2/3 of the way from the orthocenter to the circumcenter,[2][3] so
Thus, if any two of these four triangle centers are known, the positions of the other two may be determined from them.
Andrew Guinand proved in 1984, as part of what is now known as Euler's triangle determination problem, that if the positions of these centers are given for an unknown triangle, then the incenter of the triangle lies within the orthocentroidal circle (the circle having the segment from the centroid to the orthocenter as its diameter). The only point inside this circle that cannot be the incenter is the nine-point center, and every other interior point of the circle is the incenter of a unique triangle.[4][5][6][7]
The distance from the nine-point center to the incenterI satisfies
The nine-point center is the circumcenter of the medial triangle of the given triangle, the circumcenter of the orthic triangle of the given triangle, and the circumcenter of the Euler triangle.[3] More generally it is the circumcenter of any triangle defined from three of the nine points defining the nine-point circle.
The nine-point center lies at the centroid of four points: the triangle's three vertices and its orthocenter.[8]
The Euler lines of the four triangles formed by an orthocentric system (a set of four points such that each is the orthocenter of the triangle with vertices at the other three points) are concurrent at the nine-point center common to all of the triangles.[9]: p.111
Of the nine points defining the nine-point circle, the three midpoints of line segments between the vertices and the orthocenter are reflections of the triangle's midpoints about its nine-point center. Thus, the nine-point center forms the center of a point reflection that maps the medial triangle to the Euler triangle, and vice versa.[3]
According to Lester's theorem, the nine-point center lies on a common circle with three other points: the two Fermat points and the circumcenter.[10]
Thus if and only if two of the vertex angles differ from each other by more than 90°, one of the barycentric coordinates is negative and so the nine-point center is outside the triangle.
Referencesedit
^ abKimberling, Clark (1994), "Central Points and Central Lines in the Plane of a Triangle", Mathematics Magazine, 67 (3): 163–187, doi:10.2307/2690608, JSTOR 2690608, MR 1573021.
^ abcdEncyclopedia of Triangle Centers, accessed 2014-10-23.
^Stern, Joseph (2007), "Euler's triangle determination problem" (PDF), Forum Geometricorum, 7: 1–9.
^Euler, Leonhard (1767), "Solutio facilis problematum quorundam geometricorum difficillimorum", Novi Commentarii Academiae Scientiarum Petropolitanae (in Latin), 11: 103–123.
^Guinand, Andrew P. (1984), "Euler lines, tritangent centers, and their triangles", American Mathematical Monthly, 91 (5): 290–300, doi:10.2307/2322671, JSTOR 2322671.
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In geometry the nine point center is a triangle center a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle It is so called because it is the center of the nine point circle a circle that passes through nine significant points of the triangle the midpoints of the three edges the feet of the three altitudes and the points halfway between the orthocenter and each of the three vertices The nine point center is listed as point X 5 in Clark Kimberling s Encyclopedia of Triangle Centers 1 2 A triangle showing its circumcircle and circumcenter black altitudes and orthocenter red and nine point circle and nine point center blue Contents 1 Properties 2 Coordinates 3 References 4 External linksProperties editThe nine point center N lies on the Euler line of its triangle at the midpoint between that triangle s orthocenter H and circumcenter O The centroid G also lies on the same line 2 3 of the way from the orthocenter to the circumcenter 2 3 so N O N H 3 N G displaystyle NO NH 3 NG nbsp Thus if any two of these four triangle centers are known the positions of the other two may be determined from them Andrew Guinand proved in 1984 as part of what is now known as Euler s triangle determination problem that if the positions of these centers are given for an unknown triangle then the incenter of the triangle lies within the orthocentroidal circle the circle having the segment from the centroid to the orthocenter as its diameter The only point inside this circle that cannot be the incenter is the nine point center and every other interior point of the circle is the incenter of a unique triangle 4 5 6 7 The distance from the nine point center to the incenter I satisfies I N lt 1 2 I O I N 1 2 R 2 r lt R 2 2 R I N O I 2 displaystyle begin aligned amp IN lt tfrac 1 2 IO amp IN tfrac 1 2 R 2r lt frac R 2 amp 2R cdot IN OI 2 end aligned nbsp where R r are the circumradius and inradius respectively The nine point center is the circumcenter of the medial triangle of the given triangle the circumcenter of the orthic triangle of the given triangle and the circumcenter of the Euler triangle 3 More generally it is the circumcenter of any triangle defined from three of the nine points defining the nine point circle The nine point center lies at the centroid of four points the triangle s three vertices and its orthocenter 8 The Euler lines of the four triangles formed by an orthocentric system a set of four points such that each is the orthocenter of the triangle with vertices at the other three points are concurrent at the nine point center common to all of the triangles 9 p 111 Of the nine points defining the nine point circle the three midpoints of line segments between the vertices and the orthocenter are reflections of the triangle s midpoints about its nine point center Thus the nine point center forms the center of a point reflection that maps the medial triangle to the Euler triangle and vice versa 3 According to Lester s theorem the nine point center lies on a common circle with three other points the two Fermat points and the circumcenter 10 The Kosnita point of a triangle a triangle center associated with Kosnita s theorem is the isogonal conjugate of the nine point center 11 Coordinates editTrilinear coordinates for the nine point center are 1 2 cos B C cos C A cos A B cos A 2 cos B cos C cos B 2 cos C cos A cos C 2 cos A cos B cos A 2 sin B sin C cos B 2 sin C sin A cos C 2 sin A sin B b c a 2 b 2 c 2 b 2 c 2 2 c a b 2 c 2 a 2 c 2 a 2 2 a b c 2 a 2 b 2 a 2 b 2 2 displaystyle begin aligned amp cos B C cos C A cos A B amp cos A 2 cos B cos C cos B 2 cos C cos A cos C 2 cos A cos B amp cos A 2 sin B sin C cos B 2 sin C sin A cos C 2 sin A sin B amp bc left a 2 b 2 c 2 b 2 c 2 2 right ca left b 2 c 2 a 2 c 2 a 2 2 right ab left c 2 a 2 b 2 a 2 b 2 2 right end aligned nbsp The barycentric coordinates of the nine point center are 2 a cos B C b cos C A c cos A B a 2 b 2 c 2 b 2 c 2 2 b 2 c 2 a 2 c 2 a 2 2 c 2 a 2 b 2 a 2 b 2 2 displaystyle begin aligned amp a cos B C b cos C A c cos A B amp a 2 b 2 c 2 b 2 c 2 2 b 2 c 2 a 2 c 2 a 2 2 c 2 a 2 b 2 a 2 b 2 2 end aligned nbsp Thus if and only if two of the vertex angles differ from each other by more than 90 one of the barycentric coordinates is negative and so the nine point center is outside the triangle References edit a b Kimberling Clark 1994 Central Points and Central Lines in the Plane of a Triangle Mathematics Magazine 67 3 163 187 doi 10 2307 2690608 JSTOR 2690608 MR 1573021 a b c d Encyclopedia of Triangle Centers accessed 2014 10 23 a b c Dekov Deko 2007 Nine point center PDF Journal of Computer Generated Euclidean Geometry Stern Joseph 2007 Euler s triangle determination problem PDF Forum Geometricorum 7 1 9 Euler Leonhard 1767 Solutio facilis problematum quorundam geometricorum difficillimorum Novi Commentarii Academiae Scientiarum Petropolitanae in Latin 11 103 123 Guinand Andrew P 1984 Euler lines tritangent centers and their triangles American Mathematical Monthly 91 5 290 300 doi 10 2307 2322671 JSTOR 2322671 Franzsen William N The distance from the incenter to the Euler line Forum Geometricorum 11 2011 231 236 http forumgeom fau edu FG2011volume11 FG201126index html The Encyclopedia of Triangle Centers credits this observation to Randy Hutson 2011 Altshiller Court Nathan College Geometry Dover Publications 2007 orig Barnes amp Noble 1952 Yiu Paul 2010 The circles of Lester Evans Parry and their generalizations Forum Geometricorum 10 175 209 MR 2868943 Rigby John 1997 Brief notes on some forgotten geometrical theorems Mathematics and Informatics Quarterly 7 156 158 External links editWeisstein Eric W Nine Point Center MathWorld Retrieved from https en wikipedia org w index php title Nine point center amp oldid 1135753209, wikipedia, wiki, book, books, library,
