Triangle Centers Maria Nogin (based on joint work with Larry Cusick) Undergraduate Mathematics Seminar California State University, Fresno September 1, 2017 Outline • Triangle Centers I Well-known centers F Center of mass F Incenter F Circumcenter F Orthocenter I Not so well-known centers (and Morley's theorem) I New centers • Better coordinate systems I Trilinear coordinates I Barycentric coordinates I So what qualifies as a triangle center? • Open problems (= possible projects) mass m mass m mass m Centroid (center of mass) C Mb Ma Centroid A Mc B Three medians in every triangle are concurrent. Centroid is the point of intersection of the three medians. mass m mass m mass m Centroid (center of mass) C Mb Ma Centroid A Mc B Three medians in every triangle are concurrent. Centroid is the point of intersection of the three medians. Centroid (center of mass) C mass m Mb Ma Centroid mass m mass m A Mc B Three medians in every triangle are concurrent. Centroid is the point of intersection of the three medians. Incenter C Incenter A B Three angle bisectors in every triangle are concurrent. Incenter is the point of intersection of the three angle bisectors. Circumcenter C Mb Ma Circumcenter A Mc B Three side perpendicular bisectors in every triangle are concurrent. Circumcenter is the point of intersection of the three side perpendicular bisectors. Orthocenter C Ha Hb Orthocenter A Hc B Three altitudes in every triangle are concurrent. Orthocenter is the point of intersection of the three altitudes. Euler Line C Ha Hb Orthocenter Mb Ma Centroid Circumcenter A Hc Mc B Euler line Theorem (Euler, 1765). In any triangle, its centroid, circumcenter, and orthocenter are collinear. Nine-point circle C Ha Hb Orthocenter Mb Ma Nine-point center A Hc Mc B The midpoints of sides, feet of altitudes, and midpoints of the line segments joining vertices with the orthocenter lie on a circle. Nine-point center is the center of this circle. Euler Line C Ha Hb Orthocenter Mb Nine-point center Ma Centroid Circumcenter A Hc Mc B Euler line The nine-point center lies on the Euler line also! It is exactly midway between the orthocenter and the circumcenter. Morley's Theorem C Q P R A B Theorem (Morley, 1899). 4P QR is equilateral. The centroid of 4P QR is called the first Morley center of 4ABC. PU, QV , RW Classical concurrencies C Q P R A B The following line segments are concurrent: AP , BQ, CR PU, QV , RW Classical concurrencies C W Q P U V R A B The following line segments are concurrent: AP , BQ, CRAU , BV , CW Classical concurrencies C W Q P U V R A B The following line segments are concurrent: AP , BQ, CR AU, BV , CW PU, QV , RW PI, QJ, RK New Concurrency I C K F G Q P I J R H A B Theorem (Cusick and Nogin, 2006). The following line segments are concurrent: AF , BG, CH New Concurrency II C K F G Q P I J R H A B Theorem (Cusick and Nogin, 2006). The following line segments are concurrent: AF , BG, CH PI, QJ, RK Trilinear Coordinates C b a da db P dc A c B Trilinear coordinates: triple (t1; t2; t3) such that t1 : t2 : t3 = da : db : dc e.g. A(1; 0; 0);B(0; 1; 0);C(0; 0; 1) Barycentric Coordinates C mass λ3 P mass λ1 mass λ2 A B Barycentric coordinates: triple (λ1; λ2; λ3) such that P is the center of mass of the system fmass λ at A, mass λ at B, mass λ at Cg, i.e. −! −! −! 1 −! 2 3 λ1 A + λ2 B + λ3 C = (λ1 + λ2 + λ3)P λ1 : λ2 : λ3 = Area(P BC) : Area(P AC) : Area(P AB) Trilinears vs. Barycentrics C b a da db P dc A c B Trilinears: t1 : t2 : t3 = da : db : dc Barycentrics: λ1 : λ2 : λ3 = Area(P BC) : Area(P AC) : Area(P AB) = ada : bdb : cdc = at1 : bt2 : ct3 Centroid (center of mass) C mass m Mb Ma Centroid mass m mass m A Mc B 1 1 1 Trilinear coordinates: a : b : c Barycentric coordinates: 1 : 1 : 1 Incenter C Incenter A B Trilinear coordinates: 1 : 1 : 1 Barycentric coordinates: a : b : c Circumcenter C Mb Ma Circumcenter A Mc B Trilinear coordinates: cos(A) : cos(B) : cos(C) Barycentric coordinates: sin(2A) : sin(2B) : sin(2C) Orthocenter C Ha Hb Orthocenter A Hc B Trilinear coordinates: sec(A) : sec(B) : sec(C) Barycentric coordinates: tan(A) : tan(B) : tan(C) What is a triangle center? C b a P A c B A point P is a triangle center if it has a trilinear representation of the form f(a; b; c): f(b; c; a): f(c; a; b) such that f(a; b; c) = f(a; c; b) (such coordinates are called homogeneous in the variables a, b, c). Open problems (= possible projects) 1. Find the trilinear or barycentric coordinates of both points of concurrency: C C A B A B 2. Are these points same as some known triangle centers? Open problems (= possible projects) 3. Find an elementary geometry proof of this concurrency: C A B 4. Any other concurrencies? Thank you!.