[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki11\/mixtilinear-incircles-of-a-triangle\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki11\/mixtilinear-incircles-of-a-triangle\/","headline":"Mixtilinear incircles of a triangle","name":"Mixtilinear incircles of a triangle","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In geometry, a mixtilinear incircle of a triangle is a circle tangent to","datePublished":"2022-07-02","dateModified":"2022-07-02","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki11\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki11\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?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:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki11\/mixtilinear-incircles-of-a-triangle\/","wordCount":8948,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In geometry, a mixtilinear incircle of a triangle is a circle tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex A{displaystyle A} is called the        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4A{displaystyle A}-mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.  A{displaystyle A}-Mixtilinear incircle of triangle        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4ABC{displaystyle ABC}Table of ContentsProof of existence and uniqueness[edit]Construction[edit]Lemma[edit]Proof[edit]Other properties[edit]Radius[edit]Relationship with points on the circumcircle[edit]Circles related to the tangency point with the circumcircle[edit]Spiral similarities[edit]Relationship between the three mixtilinear incircles[edit]Lines joining vertices and mixtilinear tangency points[edit]Radical center[edit]References[edit]Proof of existence and uniqueness[edit]The A{displaystyle A}-excircle of triangle ABC{displaystyle ABC} is unique. Let \u03a6{displaystyle Phi } be a transformation defined by the composition of an inversion centered at A{displaystyle A} with radius AB\u22c5AC{displaystyle {sqrt {ABcdot AC}}} and a reflection with respect to the angle bisector on A{displaystyle A}. Since inversion and reflection are bijective and preserve touching points, then \u03a6{displaystyle Phi } does as well. Then, the image of the A{displaystyle A}-excircle under \u03a6{displaystyle Phi } is a circle internally tangent to sides AB,AC{displaystyle AB,AC} and the circumcircle of ABC{displaystyle ABC}, that is, the A{displaystyle A}-mixtilinear incircle. Therefore, the A{displaystyle A}-mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to B{displaystyle B} and C{displaystyle C}.[1]Construction[edit]  The hexagon XCABYTA{displaystyle XCABYT_{A}} and the intersections D,I,E{displaystyle D,I,E} of its 3 pairs of opposite sides.The A{displaystyle A}-mixtilinear incircle can be constructed with the following sequence of steps.[2]Draw the incenter I{displaystyle I} by intersecting angle bisectors.Draw a line through I{displaystyle I} perpendicular to the line AI{displaystyle AI}, touching lines AB{displaystyle AB} and AC{displaystyle AC} at points D{displaystyle D} and E{displaystyle E} respectively. These are the tangent points of the mixtilinear circle.Draw perpendiculars to AB{displaystyle AB} and AC{displaystyle AC} through points D{displaystyle D} and E{displaystyle E} respectively, and intersect them in OA{displaystyle O_{A}}. OA{displaystyle O_{A}} is the center of the circle, so a circle with center OA{displaystyle O_{A}} and radius  OAE{displaystyle O_{A}E} is the mixtilinear incircleThis construction is possible because of the following fact:Lemma[edit]The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.Proof[edit]Let \u0393{displaystyle Gamma } be the circumcircle of triangle ABC{displaystyle ABC} and TA{displaystyle T_{A}} be the tangency point of the A{displaystyle A}-mixtilinear incircle \u03a9A{displaystyle Omega _{A}} and \u0393{displaystyle Gamma }. Let X\u2260TA{displaystyle Xneq T_{A}} be the intersection of line TAD{displaystyle T_{A}D} with \u0393{displaystyle Gamma } and Y\u2260TA{displaystyle Yneq T_{A}} be the intersection of line TAE{displaystyle T_{A}E} with \u0393{displaystyle Gamma }. Homothety with center on TA{displaystyle T_{A}} between \u03a9A{displaystyle Omega _{A}} and \u0393{displaystyle Gamma } implies that X,Y{displaystyle X,Y} are the midpoints of \u0393{displaystyle Gamma } arcs AB{displaystyle AB} and AC{displaystyle AC} respectively. The inscribed angle theorem implies that X,I,C{displaystyle X,I,C} and Y,I,B{displaystyle Y,I,B}  are triples of collinear points. Pascal’s theorem on hexagon XCABYTA{displaystyle XCABYT_{A}} inscribed in \u0393{displaystyle Gamma } implies that D,I,E{displaystyle D,I,E} are collinear. Since the angles \u2220DAI{displaystyle angle {DAI}} and \u2220IAE{displaystyle angle {IAE}} are equal, it follows that I{displaystyle I} is the midpoint of segment DE{displaystyle DE}.[1]Other properties[edit]Radius[edit]The following formula relates the radius r{displaystyle r} of the incircle and the radius \u03c1A{displaystyle rho _{A}} of the A{displaystyle A}-mixtilinear incircle of a triangle ABC{displaystyle ABC}:r=\u03c1A\u22c5cos2\u2061\u03b12{displaystyle r=rho _{A}cdot cos ^{2}{frac {alpha }{2}}} where \u03b1{displaystyle alpha } is the magnitude of the angle at A{displaystyle A}.[3]Relationship with points on the circumcircle[edit]Circles related to the tangency point with the circumcircle[edit]TABDI{displaystyle T_{A}BDI} and TACEI{displaystyle T_{A}CEI} are cyclic quadrilaterals.[4]Spiral similarities[edit]TA{displaystyle T_{A}} is the center of a spiral similarity that maps B,I{displaystyle B,I} to I,C{displaystyle I,C} respectively.[1]Relationship between the three mixtilinear incircles[edit]Lines joining vertices and mixtilinear tangency points[edit]The three lines joining a vertex to the point of contact of the circumcircle with the corresponding mixtilinear incircle meet at the external center of similitude of the incircle and circumcircle.[3] The Online Encyclopedia of Triangle Centers lists this point as X(56).[6] It is defined by trilinear coordinates ac+a\u2212b:bc+a\u2212b:ca+b\u2212c{displaystyle {frac {a}{c+a-b}}:{frac {b}{c+a-b}}:{frac {c}{a+b-c}}} and barycentric coordinates a2c+a\u2212b:b2c+a\u2212b:c2a+b\u2212c{displaystyle {frac {a^{2}}{c+a-b}}:{frac {b^{2}}{c+a-b}}:{frac {c^{2}}{a+b-c}}} .Radical center[edit]The radical center of the three mixtilinear incircles is the point J{displaystyle J} which divides OI{displaystyle OI} in the ratio OJ:JI=2R:\u2212r{displaystyle OJ:JI=2R:-r}where I,r,O,R{displaystyle I,r,O,R} are the incenter, inradius, circumcenter and circumradius respectively.[5]References[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\/wiki11\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki11\/mixtilinear-incircles-of-a-triangle\/#breadcrumbitem","name":"Mixtilinear incircles of a triangle"}}]}]