X3071
INTERSECTION OF X4 X_6 AND X5 X_371
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 489 }: Central Line L(2, 489) { 4, 6 }: Van Aubel Line { 5, 371 }: Central Line L(5, 371) { 30, 372 }: Central Line L(30, 372) { 3, 486 }: Central Line L(3, 486)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_489 = CEVAPOINT OF X(20) AND X(487) (0.4404, 0.3605) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5652, 0.2097) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7148, 0.4568) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_486 = INNER VECTEN POINT (-0.0810, -0.0170) X_3071 = INTERSECTION OF X(4)X(6) AND X(5)X(371) (0.5859, 0.1197)
A
B
C
Ma
Mb
Mc
X2
X489
X4
X6
X5
X371
X30
X372
X3
X486
X3071
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 489 }: Central Line L(2, 489) { 4, 6 }: Van Aubel Line { 5, 371 }: Central Line L(5, 371) { 30, 372 }: Central Line L(30, 372) { 3, 486 }: Central Line L(3, 486)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) M_a = Midpoint M_a of BC (0.8200, 0.2400) M_b = Midpoint M_b of CA (0.3200, 0.2400) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5467, 0.1600) X_489 = CEVAPOINT OF X(20) AND X(487) (0.3600, 0.4431) X_4 = ORTHOCENTER (0.6400, 0.4800) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5946, 0.1622) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7692, 0.4615) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_486 = INNER VECTEN POINT (2.3200, 0.2400) X_3071 = INTERSECTION OF X(4)X(6) AND X(5)X(371) (0.6400, 0.0185)
A
B
C
Ma
Mb
Mc
X2
X489
X4
X6
X5
X371
X30
X372
X3
X486
X3071
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 489 }: Central Line L(2, 489) { 4, 6 }: Van Aubel Line { 5, 371 }: Central Line L(5, 371) { 30, 372 }: Central Line L(30, 372) { 3, 486 }: Central Line L(3, 486)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) M_a = Midpoint M_a of BC (0.8090, 0.2429) M_b = Midpoint M_b of CA (0.3090, 0.2429) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5393, 0.1620) X_489 = CEVAPOINT OF X(20) AND X(487) (0.3820, 0.4592) X_4 = ORTHOCENTER (0.6180, 0.4859) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5794, 0.1635) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7296, 0.4725) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_486 = INNER VECTEN POINT (2.6471, 0.2429) X_3071 = INTERSECTION OF X(4)X(6) AND X(5)X(371) (0.6180, 0.0134)
A
B
C
Ma
Mb
Mc
X2
X489
X4
X6
X5
X371
X30
X372
X3
X486
X3071
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 489 }: Central Line L(2, 489) { 4, 6 }: Van Aubel Line { 5, 371 }: Central Line L(5, 371) { 30, 372 }: Central Line L(30, 372) { 3, 486 }: Central Line L(3, 486)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) M_a = Midpoint M_a of BC (0.8166, 0.1400) M_b = Midpoint M_b of CA (0.3166, 0.1400) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5444, 0.0934) X_489 = CEVAPOINT OF X(20) AND X(487) (0.2945, 0.6060) X_4 = ORTHOCENTER (0.6331, 0.8294) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.6182, 0.0560) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7352, 0.3832) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_486 = INNER VECTEN POINT (0.8351, -0.0536) X_3071 = INTERSECTION OF X(4)X(6) AND X(5)X(371) (0.6693, -0.1630)
A
B
C
Ma
Mb
Mc
X2
X489
X4
X6
X5
X371
X30
X372
X3
X486
X3071
Kimberling (6, 9, 13) | AB = 1
X3071 (INTERSECTION OF X4 X_6 AND X5 X_371 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sin A - 2 \cos B \cos C : \sin B - 2 \cos C \cos A : \sin C - 2 \cos A \cos B\)
Barycentrics: \((\sin A)(\sin A - 2 \cos B \cos C) : (\sin B)(\sin B - 2 \cos C \cos A) : (\sin C)(\sin C - 2 \cos A \cos B)\)
Search Value (6, 9, 13): \(4.292325038685\)
X3071
▸ Incident Central Lines13 lines
{2 489 3 486 4 6 5 371 11 2067 12 2066 20 1152 30 372 488 591 641 2482 1124 1478 1335 1479 1836 2362
▸ Functional Relations6 relations
Notes
Let A'B'C' be the outer Vecten triangle and A″B″C″ the inner Vecten triangle. Let A* be the isogonal conjugate , wrt A″B″C″ , of A' , and define B* and C* cyclically. The lines A″A* , B″B* , C″C* concur in X3071 Randy Hutson , July 11, 2019)
