X149

REFLECTION OF X20 IN X104

X149 is the {X11,X100}-harmonic conjugate of X2. For a list of harmonic conjugates, click Tables at the top of this page.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 20, 104 }: Central Line L(20, 104) { 2, 11 }: Central Line L(2, 11) { 8, 80 }: Central Line L(8, 80) { 4, 145 }: Central Line L(4, 145) { 404, 496 }: Central Line L(404, 496) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_2 = CENTROID (0.5374, 0.2000) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_8 = NAGEL POINT (0.4694, 0.1333) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_4 = ORTHOCENTER (0.6122, 0.3958) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_404 = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21) (0.5538, 0.2429) X_496 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(495) (0.5843, 0.2201) X_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) A B C X20 X104 X2 X11 X8 X80 X4 X145 X404 X496 X149 (a, b, c) = (5, 6, 7) | AB = 1

X149 (REFLECTION OF X20 IN X104) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[b^{3} + c^{3} - a^{3} + (a^{2} - bc)(b + c) + a(bc - b^{2} - c^{2})]\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = b^{3} + c^{3} - a^{3} + (a^{2} - bc)(b + c) + a(bc - b^{2} - c^{2})\)
Search Value (6, 9, 13): \(2.684434107515\)

X149 lies on the anticomplementary circle.

▸ Incident Central Lines7 lines

{2, 11}, {4, 145}, {8, 80}, {20, 104}, {151, 962}, {377, 1058}, {404, 496}

▸ Functional Relations8 relations

Notes

Let A' be the reflection of the midpoint of segment BC in X11, and define B' and C' cyclically. The lines AA', BB', CC' concur in X149. (Randy Hutson, 9/23/2011)