Triangle Centers. Maria Nogin. (based on joint work with Larry Cusick)
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1 Triangle enters Maria Nogin (based on joint work with Larry usick) Undergraduate Mathematics Seminar alifornia State University, Fresno September 1, 2017
2 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)
3 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.
4 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.
5 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.
6 Incenter Incenter Three angle bisectors in every triangle are concurrent. Incenter is the point of intersection of the three angle bisectors.
7 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.
8 Orthocenter Hb H a Orthocenter H c Three altitudes in every triangle are concurrent. Orthocenter is the point of intersection of the three altitudes.
9 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.
10 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.
11 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.
12 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.
13 lassical concurrencies Q P R The following line segments are concurrent: P, Q, R
14 lassical concurrencies Q W P U R V The following line segments are concurrent: P, Q, R U, V, W
15 lassical concurrencies Q W P U R V The following line segments are concurrent: P, Q, R U, V, W P U, QV, RW
16 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
17 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
18 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)
19 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 )
20 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
21 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
22 Incenter Incenter Trilinear coordinates: 1 : 1 : 1 arycentric coordinates: a : b : c
23 ircumcenter M b M a ircumcenter M c Trilinear coordinates: cos() : cos() : cos() arycentric coordinates: sin(2) : sin(2) : sin(2)
24 Orthocenter Hb H a Orthocenter H c Trilinear coordinates: sec() : sec() : sec() arycentric coordinates: tan() : tan() : tan()
25 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).
26 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?
27 Open problems (= possible projects) 3. Find an elementary geometry proof of this concurrency: 4. ny other concurrencies?
28 Thank you!
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