X3532
INTERSECTION OF LINES X(6,1204) AND X(64,1620)
Let
DEF be the
cevian triangle of
X69 and
A' the center of the circle that passes through
D and through points where the line
EF cuts to the circumcircle, and define
B' and
C' cyclically. The lines
AA' ,
BB' ,
CC' concur in
X3532 . (
Angel Montesdeoca , October 30, 2019)
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) { 72, 165 }: Central Line L(72, 165) { 68, 550 }: Central Line L(68, 550) { 4, 1192 }: Central Line L(4, 1192) { 6, 1204 }: Central Line L(6, 1204) { 378, 1173 }: Central Line L(378, 1173) { 74, 1498 }: Central Line L(74, 1498)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5483, 0.1819) X_1152 = ISOGONAL CONJUGATE OF X(1132) (-0.9520, -2.2961) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4367, 0.0733) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_550 = MIDPOINT OF X(3) AND X(20) (0.4439, -0.0448) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1192 = 2nd SARAGOSSA POINT OF X(64) (0.6404, 0.3340) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_1204 = 3rd SARAGOSSA POINT OF X(64) (0.6497, 0.1923) X_378 = REFLECTION OF X(22) IN X(3) (0.5857, 0.3262) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6043, 0.2952) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_1498 = REFLECTION OF X(64) IN X(3) (0.4421, -1.3902) X_3532 = INTERSECTION OF LINES X(6,1204) AND X(64,1620) (0.8270, -0.0750)
A
B
C
X1151
X1152
X72
X165
X68
X550
X4
X1192
X6
X1204
X378
X1173
X74
X1498
X3532
(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) { 72, 165 }: Central Line L(72, 165) { 68, 550 }: Central Line L(68, 550) { 4, 1192 }: Van Aubel Line { 6, 1204 }: Van Aubel Line { 378, 1173 }: Van Aubel Line { 74, 1498 }: Central Line L(74, 1498)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5714, 0.1224) X_1152 = ISOGONAL CONJUGATE OF X(1132) (4.0000, 6.0000) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4000, 0.0000) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4667, -0.0667) X_68 = PRASOLOV POINT (-0.0283, 0.1600) X_550 = MIDPOINT OF X(3) AND X(20) (0.4300, -0.2400) X_4 = ORTHOCENTER (0.6400, 0.4800) X_1192 = 2nd SARAGOSSA POINT OF X(64) (0.6400, 0.2400) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_1204 = 3rd SARAGOSSA POINT OF X(64) (0.6400, 0.0000) X_378 = REFLECTION OF X(22) IN X(3) (0.6400, 0.4800) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6400, 0.2880) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6400, -0.4800) X_1498 = REFLECTION OF X(64) IN X(3) (0.5467, -0.4150) X_3532 = INTERSECTION OF LINES X(6,1204) AND X(64,1620) (0.6400, -0.2400)
A
B
C
X1151
X1152
X72
X165
X68
X550
X4
X1192
X6
X1204
X378
X1173
X74
X1498
X3532
(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) { 72, 165 }: Central Line L(72, 165) { 68, 550 }: Central Line L(68, 550) { 4, 1192 }: Van Aubel Line { 6, 1204 }: Van Aubel Line { 378, 1173 }: Van Aubel Line { 74, 1498 }: Central Line L(74, 1498)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5599, 0.1232) X_1152 = ISOGONAL CONJUGATE OF X(1132) (4.6762, 8.5953) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4159, 0.0000) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4720, -0.0674) X_68 = PRASOLOV POINT (-0.0357, 0.1620) X_550 = MIDPOINT OF X(3) AND X(20) (0.4410, -0.2429) X_4 = ORTHOCENTER (0.6180, 0.4859) X_1192 = 2nd SARAGOSSA POINT OF X(64) (0.6180, 0.2429) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_1204 = 3rd SARAGOSSA POINT OF X(64) (0.6180, 0.0000) X_378 = REFLECTION OF X(22) IN X(3) (0.6180, 0.4859) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6180, 0.2915) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6180, -0.4859) X_1498 = REFLECTION OF X(64) IN X(3) (0.5393, -0.4130) X_3532 = INTERSECTION OF LINES X(6,1204) AND X(64,1620) (0.6180, -0.2429)
A
B
C
X1151
X1152
X72
X165
X68
X550
X4
X1192
X6
X1204
X378
X1173
X74
X1498
X3532
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) { 72, 165 }: Central Line L(72, 165) { 68, 550 }: Central Line L(68, 550) { 4, 1192 }: Central Line L(4, 1192) { 6, 1204 }: Central Line L(6, 1204) { 378, 1173 }: Central Line L(378, 1173) { 74, 1498 }: Central Line L(74, 1498)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5947, -0.0098) X_1152 = ISOGONAL CONJUGATE OF X(1132) (0.9654, 1.0273) X_72 = ISOGONAL CONJUGATE OF X(28) (0.3761, -0.0722) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4615, -0.4096) X_68 = PRASOLOV POINT (0.7351, 0.0696) X_550 = MIDPOINT OF X(3) AND X(20) (0.4334, -0.8267) X_4 = ORTHOCENTER (0.6331, 0.8294) X_1192 = 2nd SARAGOSSA POINT OF X(64) (0.5499, -0.1351) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_1204 = 3rd SARAGOSSA POINT OF X(64) (0.5298, -0.4963) X_378 = REFLECTION OF X(22) IN X(3) (0.1672, -3.0347) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6333, 0.2782) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.5025, -0.8451) X_1498 = REFLECTION OF X(64) IN X(3) (0.5882, 2.0624) X_3532 = INTERSECTION OF LINES X(6,1204) AND X(64,1620) (0.5098, -0.5999)
A
B
C
X1151
X1152
X72
X165
X68
X550
X4
X1192
X6
X1204
X378
X1173
X74
X1498
X3532
Kimberling (6, 9, 13) | AB = 1
X3532 (INTERSECTION OF LINES X(6,1204) AND X(64,1620) ) 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 \(f(A,B,C) = 1/(4 \cos A - 3 \sin B \sin C)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(10.066330179103\)
X3532
▸ Incident Central Lines11 lines
{4 1192 6 1204 64 1620 68 550 71 3207 72 165 74 1498 265 1657 378 1173 1181 3431 3357 3426
▸ Functional Relations5 relations
