X3049
INTERSECTION X6 X_523∩X520 X_647
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) { 2451, 6 }: Central Line L(2451, 6) { 6, 523 }: Central Line L(6, 523) { 520, 647 }: Central Line L(520, 647) { 669, 688 }: Central Line L(669, 688) { 512, 1692 }: Central Line L(512, 1692)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2451 = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(1316) (-1.4168, 1.0380) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_520 = ISOGONAL CONJUGATE OF X(107) (-0.0490, 0.2227) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (4.2063, -0.5653) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-4.3514, 4.6163) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0103, -0.1475) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_1692 = RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE (0.7995, 0.5968) X_3049 = INTERSECTION X(6)X(523)∩X(520)X(647) (2.6168, -0.5036)
A
B
C
X2451
X6
X523
X520
X647
X669
X688
X512
X1692
X3049
(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) { 2451, 6 }: Central Line L(2451, 6) { 6, 523 }: Central Line L(6, 523) { 520, 647 }: Central Line L(520, 647) { 669, 688 }: Central Line L(669, 688) { 512, 1692 }: Central Line L(512, 1692)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2451 = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(1316) (-0.1829, 0.4800) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0987, -0.0010) X_520 = ISOGONAL CONJUGATE OF X(107) (-0.0283, 0.1600) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (2.2857, 0.0000) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-1.2857, 2.0833) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0049, -0.1873) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) X_1692 = RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE (0.7445, 0.4191) X_3049 = INTERSECTION X(6)X(523)∩X(520)X(647) (1.4629, 0.0000)
A
B
C
X2451
X6
X523
X520
X647
X669
X688
X512
X1692
X3049
(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) { 2451, 6 }: Central Line L(2451, 6) { 6, 523 }: Central Line L(6, 523) { 520, 647 }: Central Line L(520, 647) { 669, 688 }: Central Line L(669, 688) { 512, 1692 }: Central Line L(512, 1692)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2451 = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(1316) (-0.3820, 0.4859) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0981, 0.0262) X_520 = ISOGONAL CONJUGATE OF X(107) (-0.0357, 0.1620) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (2.6180, 0.0000) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-1.6180, 2.0582) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0245, -0.1466) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) X_1692 = RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE (0.7135, 0.4395) X_3049 = INTERSECTION X(6)X(523)∩X(520)X(647) (1.6180, 0.0000)
A
B
C
X2451
X6
X523
X520
X647
X669
X688
X512
X1692
X3049
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) { 2451, 6 }: Central Line L(2451, 6) { 6, 523 }: Central Line L(6, 523) { 520, 647 }: Central Line L(520, 647) { 669, 688 }: Central Line L(669, 688) { 512, 1692 }: Central Line L(512, 1692)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2451 = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(1316) (0.3154, 0.2067) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_523 = ISOGONAL CONJUGATE OF X(110) (1.1957, 0.0148) X_520 = ISOGONAL CONJUGATE OF X(107) (-0.1112, 0.1181) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (1.5059, 0.1051) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-0.1118, 0.6834) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.1304, -0.2016) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) X_1692 = RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE (0.6830, 0.2373) X_3049 = INTERSECTION X(6)X(523)∩X(520)X(647) (0.9993, 0.1243)
A
B
C
X2451
X6
X523
X520
X647
X669
X688
X512
X1692
X3049
Kimberling (6, 9, 13) | AB = 1
X3049 (INTERSECTION X6 X_523∩X520 X_647 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a^{2}[b \sin(A - B) - c \sin(A - C)] : :\)
Search Value (6, 9, 13): \(-1.278089731994\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines9 lines
{6 523 112 2713 250 2715 421 2501 512 1692 520 647 669 688 924 2485 1510 2492
▸ Functional Relations13 relations
X3049 i and Xj for these (i,j): (6 3050 2451 3288
X3049 X2451 X6
X3049 isogonal conjugate of X6331
X3049 i -Ceva conjugate of Xj for these (i,j): (2623 512 2715 237
X3049 cevapoint of X647 X2524
X3049 crosspoint of Xi and Xj for these (i,j): (6 1576 512 647
X3049 crosssum of Xi and Xj for these (i,j): (2 950 4 2489 99 648 427 2501 647 1899
X3049 crossdifference of every pair of points on line X4 X69
X3049 isotomic conjugate of X647
X3049 39201
X3049 A , B , C ,X3 X25
X3049
X3049
Notes
Let L be the isotomic conjugate of the polar conjugate of the Lemoine axis (i.e., line X520 X647 X512 X1692 X3049 Randy Hutson , June 7, 2019)
