X2144

X2-EIGENTRANSFORM OF X238

Suppose U =\(u : v : w\) and P =\(p : q : r\) are triangle centers. The P-eigentransform of U, denoted by ET(U,P), is the point given by first trilinear
< 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) { 1, 2111 }: Central Line L(1, 2111) { 2, 2113 }: Central Line L(2, 2113) { 6, 2109 }: Central Line L(6, 2109) { 2054, 2107 }: Central Line L(2054, 2107) { 2053, 2115 }: Central Line L(2053, 2115) { 238, 2145 }: Central Line L(238, 2145) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_2111 = -ISOCONJUGATE OF X(2110) (-0.1997, 0.7497) X_2 = CENTROID (0.5374, 0.2000) X_2113 = -ISOCONJUGATE OF X(2112) (-0.3440, 0.6801) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2109 = -ISOCONJUGATE OF X(2108) (-0.0481, 0.7903) X_2054 = -ISOCONJUGATE OF X(1931) (0.9934, -0.0217) X_2107 = -ISOCONJUGATE OF X(2106) (1.0304, 0.0996) X_2053 = -ISOCONJUGATE OF X(1423) (0.4486, 0.1337) X_2115 = -ISOCONJUGATE OF X(2114) (-0.5439, 0.4629) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_2145 = -ISOCONJUGATE OF X(2144) (1.7940, 1.8985) X_2144 = -EIGENTRANSFORM OF X(238) (0.9864, -0.0446) A B C X1 X2111 X2 X2113 X6 X2109 X2054 X2107 X2053 X2115 X238 X2145 X2144 (a, b, c) = (5, 6, 7) | AB = 1

X2144 (X2-EIGENTRANSFORM OF X238) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(u,v,w) : f(v,w,u) : f(w,u,v)\), where f is as given just before X2120 and \(u : v : w\) = X2382
Barycentrics: \(af(u,v,w) : bf(v,w,u) : cf(w,u,v)\)
Search Value (6, 9, 13): \(0.135245145763\)

X2144 lies on the 2nd equal-areas cubic (K155).

▸ Incident Central Lines6 lines

{1, 2111}, {2, 2113}, {6, 2109}, {238, 2145}, {2053, 2115}, {2054, 2107}

▸ Functional Relations3 relations

Notes

\(qrvw(pqu^{2}\)\(v^{2}\) + \(pru^{2}\)\(w^{2}\) - \(qrv^{2}\)\(w^{2})\).

Thus, ET(U) = ET(U,X1), and, extending the on-cubic property, ET(U,P) lies on the cubic Z(U,P) given by

\(upx(qy^{2}\) - \(rz^{2})\) + \(vqy(rz^{2}\) - \(pu^{2})\) + \(wrz(px^{2}\) - \(qv^{2})\) = 0.