X3257
ISOGONAL CONJUGATE OF X1635
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) { 44, 88 }: Central Line L(44, 88) { 106, 238 }: Central Line L(106, 238) { 100, 513 }: Central Line L(100, 513) { 527, 666 }: Central Line L(527, 666) { 320, 908 }: Central Line L(320, 908) { 662, 1019 }: Central Line L(662, 1019)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1635 = TRIPOLAR CENTROID OF X(1) (2.1429, 2.0996) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8698, 0.6836) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.0982, 0.1428) X_320 = ISOTOMIC CONJUGATE OF X(80) (-2.6735, -3.5993) X_908 = POINT ACUBENS (-0.8367, -1.7996) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.1811, 0.4951) X_1019 = -CEVA CONJUGATE OF X(1) (-29.0952, 18.1962) X_3257 = ISOGONAL CONJUGATE OF X(1635) (1.0000, -0.0000)
A
B
C
X1635
X44
X88
X106
X238
X100
X513
X527
X666
X320
X908
X662
X1019
X3257
(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) { 44, 88 }: Central Line L(44, 88) { 106, 238 }: Central Line L(106, 238) { 100, 513 }: Central Line L(100, 513) { 527, 666 }: Central Line L(527, 666) { 320, 908 }: Central Line L(320, 908) { 662, 1019 }: Central Line L(662, 1019)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1635 = TRIPOLAR CENTROID OF X(1) (1.6000, 1.2000) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_238 = -LINE CONJUGATE OF X(37) (1.2667, 0.8667) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8881, 0.6226) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.1272, 0.1723) X_320 = ISOTOMIC CONJUGATE OF X(80) (-1.5600, -1.9200) X_908 = POINT ACUBENS (-0.2800, -0.9600) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.2597, 0.4299) X_1019 = -CEVA CONJUGATE OF X(1) (-11.4000, 7.2000) X_3257 = ISOGONAL CONJUGATE OF X(1635) (1.0000, -0.0000)
A
B
C
X1635
X44
X88
X106
X238
X100
X513
X527
X666
X320
X908
X662
X1019
X3257
(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) { 44, 88 }: Central Line L(44, 88) { 106, 238 }: Central Line L(106, 238) { 100, 513 }: Central Line L(100, 513) { 527, 666 }: Central Line L(527, 666) { 320, 908 }: Central Line L(320, 908) { 662, 1019 }: Central Line L(662, 1019)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1635 = TRIPOLAR CENTROID OF X(1) (1.8360, 1.1813) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9710, -0.0490) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) X_238 = -LINE CONJUGATE OF X(37) (1.2021, 0.9450) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8397, 0.6523) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.1180, 0.2429) X_320 = ISOTOMIC CONJUGATE OF X(80) (-1.4042, -2.1471) X_908 = POINT ACUBENS (-0.1876, -1.0248) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.2938, 0.4492) X_1019 = -CEVA CONJUGATE OF X(1) (-13.7166, 7.1503) X_3257 = ISOGONAL CONJUGATE OF X(1635) (1.0290, 0.0975)
A
B
C
X1635
X44
X88
X106
X238
X100
X513
X527
X666
X320
X908
X662
X1019
X3257
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) { 44, 88 }: Central Line L(44, 88) { 106, 238 }: Central Line L(106, 238) { 100, 513 }: Central Line L(100, 513) { 527, 666 }: Central Line L(527, 666) { 320, 908 }: Central Line L(320, 908) { 662, 1019 }: Central Line L(662, 1019)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1635 = TRIPOLAR CENTROID OF X(1) (1.3278, 0.4768) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9802, -0.0393) X_106 = Λ(INCENTER, CENTROID) (1.0294, -0.0621) X_238 = -LINE CONJUGATE OF X(37) (0.9397, 0.4041) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.9260, 0.6270) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.0867, 0.1883) X_320 = ISOTOMIC CONJUGATE OF X(80) (-0.5706, -0.8023) X_908 = POINT ACUBENS (0.2165, -0.3651) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.3982, 0.2796) X_1019 = -CEVA CONJUGATE OF X(1) (-4.4509, 1.9411) X_3257 = ISOGONAL CONJUGATE OF X(1635) (1.0036, 0.0721)
A
B
C
X1635
X44
X88
X106
X238
X100
X513
X527
X666
X320
X908
X662
X1019
X3257
Kimberling (6, 9, 13) | AB = 1
X3257 (ISOGONAL CONJUGATE OF X1635 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/((b - c)(b + c - 2a)) : :\)
Search Value (6, 9, 13): \(-0.773518850939\)
▸ Incident Central Lines9 lines
{44 88 100 513 106 238 320 908 527 666 658 1275 662 1019 758 1168 897 1757
▸ Functional Relations9 relations
X3257 i in Xj for these (i,j): (320 908 3218 44
X3257 X1635
X3257 cevapoint of Xi and Xj for these (i,j): (1 1635 88 1022 100 1023 649 1149
X3257 i -cross conjugate of Xj for these (i,j): (44 765 661 1168 1022 88 1023 100 1635 1 1769 75
X3257 crossdifference of every pair of points on the line X2087 X3251
X3257 isotomic conjugate of X3762
X3257 A , B , C ,PU(28)}
X3257 trilinear polars of P(28) and U(28)
X3257 trilinear pole of line X1 X88
Notes
Let P and Q be the intersections of line BC and circle {X3 A' be the circumcenter of triangle PQ X3 B' and C' cyclically. The lines AA' , BB' , CC' concur in X3257 Randy Hutson , December 26, 2015)
Let A'B'C' and A″B″C″ be the Ursa-minor and Ursa-major triangles, resp. Let A* be the reflection of A' in line B″C″ , and define B* and C* cyclically. Let A* * be the reflection of A″ in line B'C' , and define B* * and C* * cyclically. Let A# = B*B* */\ C*C* *, and define B# and C# cyclically. The lines AA#, BB#, CC# concur in X3257
