Triangle enters Maria Nogin (based on joint work with Larry usick) Undergraduate Mathematics Seminar alifornia State University, Fresno September 1, 2017
Outline Triangle enters Well-known centers enter of mass Incenter ircumcenter Orthocenter Not so well-known centers (and Morley s theorem) New centers etter coordinate systems Trilinear coordinates arycentric coordinates So what qualifies as a triangle center? Open problems (= possible projects)
entroid (center of mass) M b entroid M a M c Three medians in every triangle are concurrent. entroid is the point of intersection of the three medians.
entroid (center of mass) M b entroid M a M c Three medians in every triangle are concurrent. entroid is the point of intersection of the three medians.
entroid (center of mass) mass m M b entroid M a mass m M c mass m Three medians in every triangle are concurrent. entroid is the point of intersection of the three medians.
Incenter Incenter Three angle bisectors in every triangle are concurrent. Incenter is the point of intersection of the three angle bisectors.
ircumcenter M b M a ircumcenter M c Three side perpendicular bisectors in every triangle are concurrent. ircumcenter is the point of intersection of the three side perpendicular bisectors.
Orthocenter Hb H a Orthocenter H c Three altitudes in every triangle are concurrent. Orthocenter is the point of intersection of the three altitudes.
Euler Line Hb H a M b Orthocenter entroid ircumcenter M a H c M c Euler line Theorem (Euler, 1765). In any triangle, its centroid, circumcenter, and orthocenter are collinear.
Nine-point circle Hb H a M b Orthocenter Nine-point center M a H c M c 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 Hb H a M b Orthocenter Nine-point center M a entroid ircumcenter H c M c 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 Q P R Theorem (Morley, 1899). P QR is equilateral. The centroid of P QR is called the first Morley center of.
lassical concurrencies Q P R The following line segments are concurrent: P, Q, R
lassical concurrencies Q W P U R V The following line segments are concurrent: P, Q, R U, V, W
lassical concurrencies Q W P U R V The following line segments are concurrent: P, Q, R U, V, W P U, QV, RW
New oncurrency I G Q K P F I R J H Theorem (usick and Nogin, 2006). The following line segments are concurrent: F, G, H
New oncurrency II G Q K P F I R H J Theorem (usick and Nogin, 2006). The following line segments are concurrent: F, G, H P I, QJ, RK
Trilinear oordinates b a d b P d c d a c Trilinear coordinates: triple (t 1, t 2, t 3 ) such that t 1 : t 2 : t 3 = d a : d b : d c e.g. (1, 0, 0), (0, 1, 0), (0, 0, 1)
arycentric oordinates mass λ 3 P mass λ 1 mass λ 2 arycentric coordinates: triple (λ 1, λ 2, λ 3 ) such that P is the center of mass of the system {mass λ 1 at, mass λ 2 at, mass λ 3 at }, i.e. λ 1 + λ2 + λ3 = (λ1 + λ 2 + λ 3 ) P λ 1 : λ 2 : λ 3 = rea(p ) : rea(p ) : rea(p )
Trilinears vs. arycentrics b a d b P d c d a c Trilinears: t 1 : t 2 : t 3 = d a : d b : d c arycentrics: λ 1 : λ 2 : λ 3 = rea(p ) : rea(p ) : rea(p ) = ad a : bd b : cd c = at 1 : bt 2 : ct 3
entroid (center of mass) mass m M b entroid M a mass m M c mass m Trilinear coordinates: 1 a : 1 b : 1 c arycentric coordinates: 1 : 1 : 1
Incenter Incenter Trilinear coordinates: 1 : 1 : 1 arycentric coordinates: a : b : c
ircumcenter M b M a ircumcenter M c Trilinear coordinates: cos() : cos() : cos() arycentric coordinates: sin(2) : sin(2) : sin(2)
Orthocenter Hb H a Orthocenter H c Trilinear coordinates: sec() : sec() : sec() arycentric coordinates: tan() : tan() : tan()
What is a triangle center? b P a c 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: 2. re these points same as some known triangle centers?
Open problems (= possible projects) 3. Find an elementary geometry proof of this concurrency: 4. ny other concurrencies?
Thank you!