X1468

X958-BETH CONJUGATE OF X958

X1468 = crosssum of X9 and X612
< 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, 21 }: Central Line L(1, 21) { 3, 42 }: Central Line L(3, 42) { 6, 41 }: Central Line L(6, 41) { 8, 171 }: Central Line L(8, 171) { 36, 386 }: Central Line L(36, 386) { 43, 404 }: Central Line L(43, 404) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_958 = INTERSECTION OF LINES X(1)X(6) AND X(2)X(12) (0.5166, 0.1682) X_1 = INCENTER (0.5714, 0.2333) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_41 = -CEVA CONJUGATE OF X(31) (0.5823, 0.2244) X_8 = NAGEL POINT (0.4694, 0.1333) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5819, 0.2525) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_404 = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21) (0.5538, 0.2429) X_1468 = -BETH CONJUGATE OF X(958) (0.6033, 0.2751) A B C X958 X1 X21 X3 X42 X6 X41 X8 X171 X36 X386 X43 X404 X1468 (a, b, c) = (5, 6, 7) | AB = 1

X1468 (X958-BETH CONJUGATE OF X958) 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 \(f(a,b,c) = (1 - \cos A)u(a,b,c)\), where \(u : v : w\) = X958
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.896837686936\)
▸ Alternate Coordinates1 form

  • Trilinears: \(a(as + bc) : b(bs + ca) : c(cs + ab)\)

▸ Incident Central Lines26 lines

{1, 21}, {3, 42}, {4, 1430}, {6, 41}, {8, 171}, {10, 750}, {36, 386}, {43, 404}, {57, 961}, {65, 603}, {75, 757}, {222, 1042}, {294, 1416}, {330, 985}, {354, 1104}, {474, 899}, {517, 601}, {518, 976}, {602, 1385}, {614, 1453}, {748, 1125}, {756, 975}, {940, 958}, {995, 1203}, {999, 1201}, {1149, 1191}

▸ Functional Relations3 relations

Notes

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius \(\sqrt{r_a^{2} + s^{2}}\), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb ∩ Lc, B' = Lc ∩ La, C' = La ∩ Lb. The lines AA', BB', CC' concur in X1468. (Randy Hutson, April 9, 2016)