X2024
RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES
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, 6 }: Brocard Axis { 230, 732 }: Central Line L(230, 732) { 1503, 2023 }: Central Line L(1503, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_732 = EVEN (- 2, 2) INFINITY POINT (0.9745, 0.5915) X_1503 = POINT ARKAB (0.4818, -0.3843) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6642, 0.1572) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6887, 0.4136)
A
B
C
X3
X6
X230
X732
X1503
X2023
X2024
(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, 6 }: Brocard Axis { 230, 732 }: Central Line L(230, 732) { 1503, 2023 }: Central Line L(1503, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_230 = -CEVA CONJUGATE OF X(114) (0.8489, 0.4191) X_732 = EVEN (- 2, 2) INFINITY POINT (1.0266, 0.4766) X_1503 = POINT ARKAB (0.5467, -0.4150) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6791, 0.1280) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6791, 0.3071)
A
B
C
X3
X6
X230
X732
X1503
X2023
X2024
(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, 6 }: Brocard Axis { 230, 732 }: Central Line L(230, 732) { 1503, 2023 }: Central Line L(1503, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_230 = -CEVA CONJUGATE OF X(114) (0.8090, 0.4395) X_732 = EVEN (- 2, 2) INFINITY POINT (0.9914, 0.5173) X_1503 = POINT ARKAB (0.5393, -0.4130) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6545, 0.1215) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6545, 0.3180)
A
B
C
X3
X6
X230
X732
X1503
X2023
X2024
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, 6 }: Brocard Axis { 230, 732 }: Central Line L(230, 732) { 1503, 2023 }: Central Line L(1503, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_230 = -CEVA CONJUGATE OF X(114) (0.7126, 0.2008) X_732 = EVEN (- 2, 2) INFINITY POINT (1.1491, 0.3477) X_1503 = POINT ARKAB (0.5683, -0.5623) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6664, 0.0713) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6624, 0.1797)
A
B
C
X3
X6
X230
X732
X1503
X2023
X2024
Kimberling (6, 9, 13) | AB = 1
X2024 (RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sin A - 3 \sin(A - 2\omega ) - 2 \cos \omega \sin(A + 3\omega ) : :\)
Search Value (6, 9, 13): \(0.806395365321\)
▸ Incident Central Lines3 lines
{3 6 230 732 1503 2023
▸ Functional Relations4 relations
X2024 i and Xj for these (i,j): (39 1692 2021 2025
X2024 X371 X372
X2024 X371 X372
X2024 35436
