X3527
ISOGONAL CONJUGATE OF X631
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) { 3, 51 }: Central Line L(3, 51) { 5, 69 }: Central Line L(5, 69) { 25, 54 }: Central Line L(25, 54) { 68, 381 }: Central Line L(68, 381) { 64, 389 }: Central Line L(64, 389) { 73, 999 }: Central Line L(73, 999)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_631 = 3/5*OG (0.5224, 0.1608) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_54 = KOSNITA POINT (0.5880, 0.2176) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5579, 1.5943) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6078, 0.2800) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_999 = MIDPOINT OF X(1) AND X(57) (0.5926, 0.2722) X_3527 = ISOGONAL CONJUGATE OF X(631) (0.6065, 0.3131)
A
B
C
X631
X3
X51
X5
X69
X25
X54
X68
X381
X64
X389
X73
X999
X3527
(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) { 3, 51 }: Central Line L(3, 51) { 5, 69 }: Central Line L(5, 69) { 25, 54 }: Van Aubel Line { 68, 381 }: Central Line L(68, 381) { 64, 389 }: Central Line L(64, 389) { 73, 999 }: Central Line L(73, 999)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_631 = 3/5*OG (0.5280, 0.0960) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6400, 0.3200) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6400, 0.4800) X_54 = KOSNITA POINT (0.6400, 0.1600) X_68 = PRASOLOV POINT (-0.0283, 0.1600) X_381 = MIDPOINT OF X(2) AND X(4) (0.5933, 0.3200) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5467, -0.4150) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6400, 0.2400) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5714, 0.0000) X_999 = MIDPOINT OF X(1) AND X(57) (0.6250, 0.2500) X_3527 = ISOGONAL CONJUGATE OF X(631) (0.6400, 0.3200)
A
B
C
X631
X3
X51
X5
X69
X25
X54
X68
X381
X64
X389
X73
X999
X3527
(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) { 3, 51 }: Central Line L(3, 51) { 5, 69 }: Central Line L(5, 69) { 25, 54 }: Van Aubel Line { 68, 381 }: Central Line L(68, 381) { 64, 389 }: Central Line L(64, 389) { 73, 999 }: Central Line L(73, 999)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_631 = 3/5*OG (0.5236, 0.0972) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6180, 0.3239) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6180, 0.4859) X_54 = KOSNITA POINT (0.6180, 0.1620) X_68 = PRASOLOV POINT (-0.0357, 0.1620) X_381 = MIDPOINT OF X(2) AND X(4) (0.5787, 0.3239) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5393, -0.4130) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6180, 0.2429) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5599, 0.0000) X_999 = MIDPOINT OF X(1) AND X(57) (0.6053, 0.2533) X_3527 = ISOGONAL CONJUGATE OF X(631) (0.6180, 0.3239)
A
B
C
X631
X3
X51
X5
X69
X25
X54
X68
X381
X64
X389
X73
X999
X3527
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) { 3, 51 }: Central Line L(3, 51) { 5, 69 }: Central Line L(5, 69) { 25, 54 }: Central Line L(25, 54) { 68, 381 }: Central Line L(68, 381) { 64, 389 }: Central Line L(64, 389) { 73, 999 }: Central Line L(73, 999)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_631 = 3/5*OG (0.5266, -0.0539) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6126, 0.2841) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.5783, 0.3748) X_54 = KOSNITA POINT (1.8800, -0.0730) X_68 = PRASOLOV POINT (0.7351, 0.0696) X_381 = MIDPOINT OF X(2) AND X(4) (0.5888, 0.4614) X_64 = ISOGONAL CONJUGATE OF X(20) (0.4118, -2.6117) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6023, 0.0114) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.6250, 0.5094) X_999 = MIDPOINT OF X(1) AND X(57) (0.6302, 0.1821) X_3527 = ISOGONAL CONJUGATE OF X(631) (0.6274, 0.3579)
A
B
C
X631
X3
X51
X5
X69
X25
X54
X68
X381
X64
X389
X73
X999
X3527
Kimberling (6, 9, 13) | AB = 1
X3527 (ISOGONAL CONJUGATE OF X631 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\) , where \(f(A,B,C) = 1/(\cos A + \sin B \sin C)\)
Search Value (6, 9, 13): \(-0.759241529748\)
▸ Alternate Coordinates2 forms
Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\) , where \(g(A,B,C) = 1/(\sec A + 2 \sec B \sec C)\)
Trilinears: \(1/(2 \cos A + \cos B \cos C) : :\)
▸ Incident Central Lines12 lines
{3 51 5 69 6 1598 24 3431 25 54 64 389 68 381 72 1482 73 999 74 1112 185 3426 1092 3066
▸ Functional Relations4 relations
Notes
Let HA be the foot of the A -altitude. Let BA and CA be the feet of perpendiculars from HA to CA and AB , resp. Let GA be the centroid of HA BA CA . Define GB and GC cyclically. The lines AGA , BGB , CGC concur in X3527 Randy Hutson , June 7, 2019)
