X3521
ISOGONAL CONJUGATE OF X3520
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) { 5, 74 }: Central Line L(5, 74) { 30, 54 }: Central Line L(30, 54) { 6, 382 }: Central Line L(6, 382) { 64, 381 }: Central Line L(64, 381) { 185, 265 }: Central Line L(185, 265) { 3, 1568 }: Central Line L(3, 1568)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1531 = ORTHOJOIN OF X(32) (0.0445, -0.5419) X_265 = REFLECTION OF X(3) IN X(125) (0.8606, -0.0500) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_54 = KOSNITA POINT (0.5880, 0.2176) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7245, 0.6894) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5579, 1.5943) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6033, 0.1643) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1568 = ORTHOJOIN OF X(800) (0.1362, -0.3019) X_3521 = ISOGONAL CONJUGATE OF X(3520) (0.5763, 0.1868)
A
B
C
X1531
X265
X5
X74
X30
X54
X6
X382
X64
X381
X185
X3
X1568
X3521
(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) { 5, 74 }: Central Line L(5, 74) { 30, 54 }: Central Line L(30, 54) { 6, 382 }: Central Line L(6, 382) { 64, 381 }: Central Line L(64, 381) { 185, 265 }: Central Line L(185, 265) { 3, 1568 }: Central Line L(3, 1568)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1531 = ORTHOJOIN OF X(32) (0.3600, 0.0000) X_265 = REFLECTION OF X(3) IN X(125) (0.7800, 0.0000) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6400, -0.4800) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_54 = KOSNITA POINT (0.6400, 0.1600) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7800, 0.9600) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5467, -0.4150) X_381 = MIDPOINT OF X(2) AND X(4) (0.5933, 0.3200) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6400, 0.0000) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1568 = ORTHOJOIN OF X(800) (0.3600, 0.0000) X_3521 = ISOGONAL CONJUGATE OF X(3520) (0.5933, 0.0000)
A
B
C
X1531
X265
X5
X74
X30
X54
X6
X382
X64
X381
X185
X3
X1568
X3521
(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) { 5, 74 }: Central Line L(5, 74) { 30, 54 }: Central Line L(30, 54) { 6, 382 }: Central Line L(6, 382) { 64, 381 }: Central Line L(64, 381) { 185, 265 }: Central Line L(185, 265) { 3, 1568 }: Central Line L(3, 1568)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1531 = ORTHOJOIN OF X(32) (0.3820, 0.0000) X_265 = REFLECTION OF X(3) IN X(125) (0.7361, 0.0000) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6180, -0.4859) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_54 = KOSNITA POINT (0.6180, 0.1620) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7361, 0.9717) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5393, -0.4130) X_381 = MIDPOINT OF X(2) AND X(4) (0.5787, 0.3239) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6180, 0.0000) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1568 = ORTHOJOIN OF X(800) (0.3820, 0.0000) X_3521 = ISOGONAL CONJUGATE OF X(3520) (0.5787, 0.0000)
A
B
C
X1531
X265
X5
X74
X30
X54
X6
X382
X64
X381
X185
X3
X1568
X3521
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) { 5, 74 }: Central Line L(5, 74) { 30, 54 }: Central Line L(30, 54) { 6, 382 }: Central Line L(6, 382) { 64, 381 }: Central Line L(64, 381) { 185, 265 }: Central Line L(185, 265) { 3, 1568 }: Central Line L(3, 1568)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1531 = ORTHOJOIN OF X(32) (0.5793, 0.6788) X_265 = REFLECTION OF X(3) IN X(125) (0.6357, 0.2589) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.5025, -0.8451) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_54 = KOSNITA POINT (1.8800, -0.0730) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7663, 1.9335) X_64 = ISOGONAL CONJUGATE OF X(20) (0.4118, -2.6117) X_381 = MIDPOINT OF X(2) AND X(4) (0.5888, 0.4614) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.5714, -0.8065) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_1568 = ORTHOJOIN OF X(800) (0.5303, 0.2721) X_3521 = ISOGONAL CONJUGATE OF X(3520) (-0.6501, -21.0551)
A
B
C
X1531
X265
X5
X74
X30
X54
X6
X382
X64
X381
X185
X3
X1568
X3521
Kimberling (6, 9, 13) | AB = 1
X3521 (ISOGONAL CONJUGATE OF X3520 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/(4 \cos A + \sec A) : :\)
Search Value (6, 9, 13): \(230.585502031644\)
▸ Incident Central Lines8 lines
{3 1568 5 74 6 382 20 3431 30 54 64 381 185 265 597 1177
▸ Functional Relations1 relation
Notes
Let A'B'C' be the orthocentroidal triangle. Let A″ be the isogonal conjugate, wrt the A-altimedial triangle, of A' . Define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X3521 Randy Hutson , November 2, 2017)
