X531
DIRECTION OF VECTOR AX + BX + CX, where X = X14
Let
A' be the nine-point center of
BC X13 , and define
B' and
C' cyclically.
A'B'C' is an equilateral triangle concyclic with
X2 and
X13 (inscribed in the 1st Hutson circle). The isogonal (and isotomic) conjugate of
X2 wrt
A'B'C' is
X531 . (
Randy Hutson , February 10, 2016)
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, 14 }: Central Line L(2, 14) { 99, 298 }: Central Line L(99, 298) { 115, 396 }: Central Line L(115, 396) { 30, 511 }: Central Line L(30, 511) { 187, 395 }: Central Line L(187, 395) { 299, 316 }: Central Line L(299, 316)
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_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_99 = STEINER POINT (0.1109, 0.4322) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4724, 0.1301) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_396 = MIDPOINT OF X(13) AND X(15) (0.5699, 0.2349) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_395 = MIDPOINT OF X(14) AND X(16) (1.3794, 1.1048) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-1.1465, -1.6097) X_316 = DROUSSENT PIVOT (-0.7850, -1.9118) X_531 = DIRECTION OF VECTOR AX + BX + CX, where X = X(14) (0.9877, -0.1764)
A
B
C
X2
X14
X99
X298
X115
X396
X30
X511
X187
X395
X299
X316
X531
(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, 14 }: Central Line L(2, 14) { 99, 298 }: Central Line L(99, 298) { 115, 396 }: Central Line L(115, 396) { 30, 511 }: Central Line L(30, 511) { 187, 395 }: Central Line L(187, 395) { 299, 316 }: Central Line L(299, 316)
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_14 = 2nd ISOGONIC CENTER (0.8701, -0.1545) X_99 = STEINER POINT (0.1511, 0.3581) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4447, 0.0726) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7445, 0.0609) X_396 = MIDPOINT OF X(13) AND X(15) (0.5976, 0.2037) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2569, -0.3367) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (0.9534, 0.7772) X_395 = MIDPOINT OF X(14) AND X(16) (1.1002, 0.6345) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-0.5604, -0.7889) X_316 = DROUSSENT PIVOT (-0.2667, -1.0744) X_531 = DIRECTION OF VECTOR AX + BX + CX, where X = X(14) (0.9589, -0.2408)
A
B
C
X2
X14
X99
X298
X115
X396
X30
X511
X187
X395
X299
X316
X531
(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, 14 }: Central Line L(2, 14) { 99, 298 }: Central Line L(99, 298) { 115, 396 }: Central Line L(115, 396) { 30, 511 }: Central Line L(30, 511) { 187, 395 }: Central Line L(187, 395) { 299, 316 }: Central Line L(299, 316)
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_14 = 2nd ISOGONIC CENTER (0.8270, -0.1871) X_99 = STEINER POINT (0.1910, 0.3931) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4539, 0.0740) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7135, 0.0464) X_396 = MIDPOINT OF X(13) AND X(15) (0.5821, 0.2059) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2881, -0.3552) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (0.9045, 0.8325) X_395 = MIDPOINT OF X(14) AND X(16) (1.0360, 0.6730) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-0.4539, -0.8602) X_316 = DROUSSENT PIVOT (-0.1910, -1.1792) X_531 = DIRECTION OF VECTOR AX + BX + CX, where X = X(14) (0.9050, -0.2818)
A
B
C
X2
X14
X99
X298
X115
X396
X30
X511
X187
X395
X299
X316
X531
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, 14 }: Central Line L(2, 14) { 99, 298 }: Central Line L(99, 298) { 115, 396 }: Central Line L(115, 396) { 30, 511 }: Central Line L(30, 511) { 187, 395 }: Central Line L(187, 395) { 299, 316 }: Central Line L(299, 316)
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_14 = 2nd ISOGONIC CENTER (0.7495, -0.2320) X_99 = STEINER POINT (0.2513, 0.2387) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4008, 0.0016) X_115 = CENTER OF KIEPERT HYPERBOLA (0.6909, 0.0207) X_396 = MIDPOINT OF X(13) AND X(15) (0.6162, 0.1392) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_511 = ISOGONAL CONJUGATE OF X(98) (0.3235, -0.5244) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (0.7344, 0.3809) X_395 = MIDPOINT OF X(14) AND X(16) (0.8091, 0.2624) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (0.0150, -0.2447) X_316 = DROUSSENT PIVOT (0.1644, -0.4818) X_531 = DIRECTION OF VECTOR AX + BX + CX, where X = X(14) (0.8942, -0.4616)
A
B
C
X2
X14
X99
X298
X115
X396
X30
X511
X187
X395
X299
X316
X531
Kimberling (6, 9, 13) | AB = 1
X531 (DIRECTION OF VECTOR AX + BX + CX, where X = X14 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((2ax - by - cz)/a : :\) , where \(x = x(A,B,C) = \csc(A - \pi /3)\) , \(y = x(B,C,A)\) , \(z = x(C,A,B)\)
X531
▸ Incident Central Lines9 lines
{2 14 13 671 30 511 99 298 115 396 148 3180 187 395 299 316 618 2482
▸ Functional Relations1 relation
