X1966
CROSSDIFFERENCE OF PU(36)
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) { 1, 75 }: Central Line L(1, 75) { 31, 561 }: Central Line L(31, 561) { 2, 893 }: Central Line L(2, 893) { 240, 811 }: Central Line L(240, 811) { 798, 812 }: Central Line L(798, 812) { 732, 894 }: Central Line L(732, 894)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1967 = TRILINEAR POLE OF PU(36) (1.0067, 0.0434) X_1 = INCENTER (0.5714, 0.2333) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_561 = ISOGONAL CONJUGATE OF 4th POWER POINT (0.4126, 0.1125) X_2 = CENTROID (0.5374, 0.2000) X_893 = -CROSS CONJUGATE OF X(292) (0.5910, 0.2480) X_240 = -LINE CONJUGATE OF X(48) (0.7457, 0.8594) X_811 = ISOGONAL CONJUGATE OF X(810) (0.4151, 0.5616) X_798 = CROSSDIFFERENCE OF X(1) AND X(75) (-13.4299, 10.5763) X_812 = ODD (- 1, 1) INFINITY POINT (0.1586, -0.2483) X_732 = EVEN (- 2, 2) INFINITY POINT (0.9745, 0.5915) X_894 = ISOGONAL CONJUGATE OF X(893) (0.5501, 0.2184) X_1966 = CROSSDIFFERENCE OF PU(36) (-93.0476, -83.6296)
A
B
C
X1967
X1
X75
X31
X561
X2
X893
X240
X811
X798
X812
X732
X894
X1966
(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) { 1, 75 }: Central Line L(1, 75) { 31, 561 }: Central Line L(31, 561) { 2, 893 }: Central Line L(2, 893) { 798, 812 }: Central Line L(798, 812) { 732, 894 }: Central Line L(732, 894) { 350, 1281 }: Central Line L(350, 1281)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1967 = TRILINEAR POLE OF PU(36) (0.9960, 0.0557) X_1 = INCENTER (0.6000, 0.2000) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4826, 0.1226) X_31 = 2nd POWER POINT (0.6667, 0.2778) X_561 = ISOGONAL CONJUGATE OF 4th POWER POINT (0.3420, 0.0633) X_2 = CENTROID (0.5467, 0.1600) X_893 = -CROSS CONJUGATE OF X(292) (0.6315, 0.2159) X_798 = CROSSDIFFERENCE OF X(1) AND X(75) (-4.9362, 4.4681) X_812 = ODD (- 1, 1) INFINITY POINT (0.1375, -0.2440) X_732 = EVEN (- 2, 2) INFINITY POINT (1.0266, 0.4766) X_894 = ISOGONAL CONJUGATE OF X(893) (0.5637, 0.1831) X_350 = -HIRST INVERSE OF X(75) (-2.3437, -1.5282) X_1281 = 3rd SHARYGIN POINT (0.3256, 0.2533) X_1966 = CROSSDIFFERENCE OF PU(36) (-29.0845, -19.3745)
A
B
C
X1967
X1
X75
X31
X561
X2
X893
X798
X812
X732
X894
X350
X1281
X1966
(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) { 1, 75 }: Central Line L(1, 75) { 31, 561 }: Central Line L(31, 561) { 2, 893 }: Central Line L(2, 893) { 798, 812 }: Central Line L(798, 812) { 732, 894 }: Central Line L(732, 894) { 350, 1281 }: Central Line L(350, 1281)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1967 = TRILINEAR POLE OF PU(36) (1.0000, 0.0000) X_1 = INCENTER (0.5841, 0.2021) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4859, 0.1249) X_31 = 2nd POWER POINT (0.6411, 0.2822) X_561 = ISOGONAL CONJUGATE OF 4th POWER POINT (0.3669, 0.0666) X_2 = CENTROID (0.5393, 0.1620) X_893 = -CROSS CONJUGATE OF X(292) (0.6106, 0.2180) X_798 = CROSSDIFFERENCE OF X(1) AND X(75) (-5.9482, 4.4191) X_812 = ODD (- 1, 1) INFINITY POINT (0.0873, -0.1934) X_732 = EVEN (- 2, 2) INFINITY POINT (0.9914, 0.5173) X_894 = ISOGONAL CONJUGATE OF X(893) (0.5538, 0.1856) X_350 = -HIRST INVERSE OF X(75) (-2.2720, -1.7862) X_1281 = 3rd SHARYGIN POINT (0.3460, 0.2720) X_1966 = CROSSDIFFERENCE OF PU(36) (0.9914, 0.5173)
A
B
C
X1967
X1
X75
X31
X561
X2
X893
X798
X812
X732
X894
X350
X1281
X1966
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) { 1, 75 }: Central Line L(1, 75) { 31, 561 }: Central Line L(31, 561) { 2, 893 }: Central Line L(2, 893) { 240, 811 }: Central Line L(240, 811) { 798, 812 }: Central Line L(798, 812) { 732, 894 }: Central Line L(732, 894)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1967 = TRILINEAR POLE OF PU(36) (0.9912, 0.0150) X_1 = INCENTER (0.6154, 0.1300) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4506, 0.0607) X_31 = 2nd POWER POINT (0.6747, 0.1958) X_561 = ISOGONAL CONJUGATE OF 4th POWER POINT (0.2571, 0.0197) X_2 = CENTROID (0.5444, 0.0934) X_893 = -CROSS CONJUGATE OF X(292) (0.6584, 0.1412) X_240 = -LINE CONJUGATE OF X(48) (0.6094, 0.1785) X_811 = ISOGONAL CONJUGATE OF X(810) (0.8957, 0.2993) X_798 = CROSSDIFFERENCE OF X(1) AND X(75) (-1.6429, 1.3237) X_812 = ODD (- 1, 1) INFINITY POINT (-0.0449, -0.1950) X_732 = EVEN (- 2, 2) INFINITY POINT (1.1491, 0.3477) X_894 = ISOGONAL CONJUGATE OF X(893) (0.5611, 0.1167) X_1966 = CROSSDIFFERENCE OF PU(36) (-32.7068, -13.8781)
A
B
C
X1967
X1
X75
X31
X561
X2
X893
X240
X811
X798
X812
X732
X894
X1966
Kimberling (6, 9, 13) | AB = 1
X1966 (CROSSDIFFERENCE OF PU(36) ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(b^{2}c^{2}(a^{4} - b^{2}c^{2}) : :\)
Search Value (6, 9, 13): \(409.289832646439\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines10 lines
{1 75 2 893 31 561 240 811 350 1281 668 1757 732 894 798 812 799 896 1821 1934
▸ Functional Relations8 relations
X1966 isogonal conjugate of X1967
X1966 isotomic conjugate of X1581
X1966 i -Ceva conjugate of Xj for these (i,j): (2 874 804 1581 1965 1821 75
X1966 i -cross conjugate of Xj for these (i,j): (698 1916 1691 699
X1966 X1580 X1582
X1966 A , B , C ,PU(35)}}
X1966 trilinear polars of P(35) and U(35)
X1966
Notes
Let A'B'C' be the 1st anti-Brocard triangle. Let A″ be the trilinear product B' *C' , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X1966 Randy Hutson , December 26, 2015)
