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 {mass λ at A, mass λ at B, mass λ at C}, i.e. −→ −→ −→ 1 −→ 2 3 λ1 A + λ2 B + λ3 C = (λ1 + λ2 + λ3)P λ1 : λ2 : λ3 = Area(PBC) : 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(PBC) : 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!