X543
DIRECTION OF VECTOR AX + BX + CX, where X = X99
5, 6, 7
3, 4, 5
Kepler
Kimberling
< 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) { 2, 99 }: Central Line L(2, 99) { 98, 376 }: Central Line L(98, 376) { 114, 381 }: Central Line L(114, 381) { 30, 511 }: Central Line L(30, 511) { 549, 1153 }: Central Line L(549, 1153) { 598, 1569 }: Central Line L(598, 1569)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_99 = STEINER POINT (0.1109, 0.4322) X_98 = TARRY POINT (0.8891, -0.2281) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_114 = KIEPERT ANTIPODE (0.3616, 0.4140) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_549 = MIDPOINT OF X(2) AND X(3) (0.5187, 0.1510) X_1153 = CENTER OF THE VAN LAMOEN CIRCLE (0.5098, 0.1559) X_598 = ISOGONAL CONJUGATE OF X(574) (0.5865, 0.2783) X_1569 = MOSES-CIRCLE ANTIPODE OF X(115) (0.4049, 0.3772) X_543 = DIRECTION OF VECTOR AX + BX + CX, where X = X(99) (0.0220, 0.4806)
A
B
C
X2
X99
X98
X376
X114
X381
X30
X511
X549
X1153
X598
X1569
X543
(a, b, c) = (5, 6, 7) | AB = 1
< 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) { 2, 99 }: Central Line L(2, 99) { 98, 376 }: Central Line L(98, 376) { 114, 381 }: Central Line L(114, 381) { 30, 511 }: Central Line L(30, 511) { 549, 1153 }: Central Line L(549, 1153) { 598, 1569 }: Central Line L(598, 1569)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_99 = STEINER POINT (0.1511, 0.3581) X_98 = TARRY POINT (0.8489, -0.3581) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4533, -0.1600) X_114 = KIEPERT ANTIPODE (0.3955, 0.4191) X_381 = MIDPOINT OF X(2) AND X(4) (0.5933, 0.3200) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2569, -0.3367) X_549 = MIDPOINT OF X(2) AND X(3) (0.5233, 0.0800) X_1153 = CENTER OF THE VAN LAMOEN CIRCLE (0.5012, 0.0911) X_598 = ISOGONAL CONJUGATE OF X(574) (0.6147, 0.2633) X_1569 = MOSES-CIRCLE ANTIPODE OF X(115) (0.4831, 0.3292) X_543 = DIRECTION OF VECTOR AX + BX + CX, where X = X(99) (0.0326, 0.4175)
A
B
C
X2
X99
X98
X376
X114
X381
X30
X511
X549
X1153
X598
X1569
X543
(a, b, c) = (3, 4, 5) | AB = 1
< 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) { 2, 99 }: Central Line L(2, 99) { 98, 376 }: Central Line L(98, 376) { 114, 381 }: Central Line L(114, 381) { 30, 511 }: Central Line L(30, 511) { 549, 1153 }: Central Line L(549, 1153) { 598, 1569 }: Central Line L(598, 1569)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_99 = STEINER POINT (0.1910, 0.3931) X_98 = TARRY POINT (0.8090, -0.3931) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4607, -0.1620) X_114 = KIEPERT ANTIPODE (0.4045, 0.4395) X_381 = MIDPOINT OF X(2) AND X(4) (0.5787, 0.3239) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2881, -0.3552) X_549 = MIDPOINT OF X(2) AND X(3) (0.5197, 0.0810) X_1153 = CENTER OF THE VAN LAMOEN CIRCLE (0.5017, 0.0929) X_598 = ISOGONAL CONJUGATE OF X(574) (0.5968, 0.2675) X_1569 = MOSES-CIRCLE ANTIPODE OF X(115) (0.4775, 0.3467) X_543 = DIRECTION OF VECTOR AX + BX + CX, where X = X(99) (0.0602, 0.4798)
A
B
C
X2
X99
X98
X376
X114
X381
X30
X511
X549
X1153
X598
X1569
X543
Kepler (1, √φ, φ) | AB = 1
< 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) { 2, 99 }: Central Line L(2, 99) { 98, 376 }: Central Line L(98, 376) { 114, 381 }: Central Line L(114, 381) { 30, 511 }: Central Line L(30, 511) { 549, 1153 }: Central Line L(549, 1153) { 598, 1569 }: Central Line L(598, 1569)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2 = CENTROID (0.5444, 0.0934) X_99 = STEINER POINT (0.2513, 0.2387) X_98 = TARRY POINT (0.7487, -0.7881) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4556, -0.6427) X_114 = KIEPERT ANTIPODE (0.4422, 0.5341) X_381 = MIDPOINT OF X(2) AND X(4) (0.5888, 0.4614) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_511 = ISOGONAL CONJUGATE OF X(98) (0.3235, -0.5244) X_549 = MIDPOINT OF X(2) AND X(3) (0.5222, -0.0907) X_1153 = CENTER OF THE VAN LAMOEN CIRCLE (0.4473, -0.0535) X_598 = ISOGONAL CONJUGATE OF X(574) (0.6210, 0.2094) X_1569 = MOSES-CIRCLE ANTIPODE OF X(115) (0.5927, 0.2234) X_543 = DIRECTION OF VECTOR AX + BX + CX, where X = X(99) (-0.0433, 0.3849)
A
B
C
X2
X99
X98
X376
X114
X381
X30
X511
X549
X1153
X598
X1569
X543
Kimberling (6, 9, 13) | AB = 1
X543 (DIRECTION OF VECTOR AX + BX + CX, where X = X99 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((2ax - by - cz)/a, x = x(a,b,c) = bc/(b^{2} - c^{2})\) , where
\(x : y : z\) =
X99 Barycentrics: \(2 a^{4} - b^{4} - c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2} + 4 b^{2} c^{2} : :\)
X543
▸ Incident Central Lines15 lines
{2 99 22 3455 25 2936 30 511 98 376 114 381 147 3543 549 1153 598 1569 626 1975 1641 2502 2421 3016 3023 3058 3027 3325 3044 3048
▸ Functional Relations9 relations
