X530
DIRECTION OF VECTOR AX + BX + CX, where X = X13
Let
A' be the nine-point center of
BC X14 , and define
B' and
C' cyclically.
A'B'C' is an equilateral triangle concyclic with
X2 and
X14 (inscribed in the 2nd Hutson circle). The isogonal (and isotomic) conjugate of
X2 wrt
A'B'C' is
X530 . (
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, 13 }: Central Line L(2, 13) { 99, 299 }: Central Line L(99, 299) { 115, 395 }: Central Line L(115, 395) { 30, 511 }: Central Line L(30, 511) { 187, 396 }: Central Line L(187, 396) { 298, 316 }: Central Line L(298, 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_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_99 = STEINER POINT (0.1109, 0.4322) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-1.1465, -1.6097) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_395 = MIDPOINT OF X(14) AND X(16) (1.3794, 1.1048) 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_396 = MIDPOINT OF X(13) AND X(15) (0.5699, 0.2349) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4724, 0.1301) X_316 = DROUSSENT PIVOT (-0.7850, -1.9118) X_530 = DIRECTION OF VECTOR AX + BX + CX, where X = X(13) (0.8451, 0.6997)
A
B
C
X2
X13
X99
X299
X115
X395
X30
X511
X187
X396
X298
X316
X530
(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, 13 }: Central Line L(2, 13) { 99, 299 }: Central Line L(99, 299) { 115, 395 }: Central Line L(115, 395) { 30, 511 }: Central Line L(30, 511) { 187, 396 }: Central Line L(187, 396) { 298, 316 }: Central Line L(298, 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_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.6188, 0.2763) X_99 = STEINER POINT (0.1511, 0.3581) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-0.5604, -0.7889) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7445, 0.0609) X_395 = MIDPOINT OF X(14) AND X(16) (1.1002, 0.6345) 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_396 = MIDPOINT OF X(13) AND X(15) (0.5976, 0.2037) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4447, 0.0726) X_316 = DROUSSENT PIVOT (-0.2667, -1.0744) X_530 = DIRECTION OF VECTOR AX + BX + CX, where X = X(13) (0.8498, 0.6486)
A
B
C
X2
X13
X99
X299
X115
X395
X30
X511
X187
X396
X298
X316
X530
(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, 13 }: Central Line L(2, 13) { 99, 299 }: Central Line L(99, 299) { 115, 395 }: Central Line L(115, 395) { 30, 511 }: Central Line L(30, 511) { 187, 396 }: Central Line L(187, 396) { 298, 316 }: Central Line L(298, 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_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.6001, 0.2799) X_99 = STEINER POINT (0.1910, 0.3931) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-0.4539, -0.8602) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7135, 0.0464) X_395 = MIDPOINT OF X(14) AND X(16) (1.0360, 0.6730) 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_396 = MIDPOINT OF X(13) AND X(15) (0.5821, 0.2059) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4539, 0.0740) X_316 = DROUSSENT PIVOT (-0.1910, -1.1792) X_530 = DIRECTION OF VECTOR AX + BX + CX, where X = X(13) (0.8024, 0.6732)
A
B
C
X2
X13
X99
X299
X115
X395
X30
X511
X187
X396
X298
X316
X530
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, 13 }: Central Line L(2, 13) { 99, 299 }: Central Line L(99, 299) { 115, 395 }: Central Line L(115, 395) { 30, 511 }: Central Line L(30, 511) { 187, 396 }: Central Line L(187, 396) { 298, 316 }: Central Line L(298, 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_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.6324, 0.2733) X_99 = STEINER POINT (0.2513, 0.2387) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (0.0150, -0.2447) X_115 = CENTER OF KIEPERT HYPERBOLA (0.6909, 0.0207) X_395 = MIDPOINT OF X(14) AND X(16) (0.8091, 0.2624) 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_396 = MIDPOINT OF X(13) AND X(15) (0.6162, 0.1392) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4008, 0.0016) X_316 = DROUSSENT PIVOT (0.1644, -0.4818) X_530 = DIRECTION OF VECTOR AX + BX + CX, where X = X(13) (0.8325, 0.6827)
A
B
C
X2
X13
X99
X299
X115
X395
X30
X511
X187
X396
X298
X316
X530
Kimberling (6, 9, 13) | AB = 1
X530 (DIRECTION OF VECTOR AX + BX + CX, where X = X13 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((2ax - by - cz)/a : :\) , where \(x = x(A,B,C) = \csc(A + \pi /3)\) , \(y = x(B,C,A)\) , \(z = x(C,A,B)\)
X530
▸ Incident Central Lines9 lines
{2 13 14 671 30 511 99 299 115 395 148 3181 187 396 298 316 619 2482
▸ Functional Relations1 relation
