[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/nine-point-center-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/nine-point-center-wikipedia\/","headline":"Nine-point center – Wikipedia","name":"Nine-point center – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Triangle center associated with the nine-point circle after-content-x4 A triangle showing its circumcircle","datePublished":"2014-02-05","dateModified":"2014-02-05","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/a4\/Ortocenter_and_circumcircle.svg\/300px-Ortocenter_and_circumcircle.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/a4\/Ortocenter_and_circumcircle.svg\/300px-Ortocenter_and_circumcircle.svg.png","height":"300","width":"300"},"url":"https:\/\/wiki.edu.vn\/en\/nine-point-center-wikipedia\/","wordCount":4397,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Triangle center associated with the nine-point circle         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4A triangle showing its circumcircle and circumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue)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]Table of ContentsProperties[edit]Coordinates[edit]References[edit]External links[edit]Properties[edit]The 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       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4|NO|=|NH|=3|NG|.{displaystyle |NO|=|NH|=3|NG|.}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|IN||IN|=|OI|2,{displaystyle {begin{aligned}&|IN|C):cos\u2061(C\u2212A):cos\u2061(A\u2212B)=cos\u2061A+2cos\u2061Bcos\u2061C:cos\u2061B+2cos\u2061Ccos\u2061A:cos\u2061C+2cos\u2061Acos\u2061B=cos\u2061A\u22122sin\u2061Bsin\u2061C:cos\u2061B\u22122sin\u2061Csin\u2061A:cos\u2061C\u22122sin\u2061Asin\u2061B=bc[a2(b2+c2)\u2212(b2\u2212c2)2]:ca[b2(c2+a2)\u2212(c2\u2212a2)2]:ab[c2(a2+b2)\u2212(a2\u2212b2)2].{displaystyle {begin{aligned}&cos(B-C):cos(C-A):cos(A-B)\\&=cos A+2cos Bcos C:cos B+2cos Ccos A:cos C+2cos Acos B\\&=cos A-2sin Bsin C:cos B-2sin Csin A:cos C-2sin Asin B\\&=bcleft[a^{2}(b^{2}+c^{2})-(b^{2}-c^{2})^{2}right]:caleft[b^{2}(c^{2}+a^{2})-(c^{2}-a^{2})^{2}right]:ableft[c^{2}(a^{2}+b^{2})-(a^{2}-b^{2})^{2}right].end{aligned}}}The barycentric coordinates of the nine-point center are[2]acos\u2061(B\u2212C):bcos\u2061(C\u2212A):ccos\u2061(A\u2212B)=a2(b2+c2)\u2212(b2\u2212c2)2:b2(c2+a2)\u2212(c2\u2212a2)2:c2(a2+b2)\u2212(a2\u2212b2)2.{displaystyle {begin{aligned}&acos(B-C):bcos(C-A):ccos(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}.end{aligned}}}Thus if and only if two of the vertex angles differ from each other by more than 90\u00b0, 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\u2013187, doi:10.2307\/2690608, JSTOR\u00a02690608, MR\u00a01573021.^ 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\u20139.^ Euler, Leonhard (1767), “Solutio facilis problematum quorundam geometricorum difficillimorum”, Novi Commentarii Academiae Scientiarum Petropolitanae (in Latin), 11: 103\u2013123.^ Guinand, Andrew P. (1984), “Euler lines, tritangent centers, and their triangles”, American Mathematical Monthly, 91 (5): 290\u2013300, doi:10.2307\/2322671, JSTOR\u00a02322671.^ 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 & Noble 1952).^ Yiu, Paul (2010), “The circles of Lester, Evans, Parry, and their generalizations”, Forum Geometricorum, 10: 175\u2013209, MR\u00a02868943.^ Rigby, John (1997), “Brief notes on some forgotten geometrical theorems”, Mathematics and Informatics Quarterly, 7: 156\u2013158.External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/nine-point-center-wikipedia\/#breadcrumbitem","name":"Nine-point center – Wikipedia"}}]}]