X3070
INTERSECTION OF X4 X_6 AND X5 X_372
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, 490 }: Central Line L(2, 490) { 4, 6 }: Van Aubel Line { 5, 372 }: Central Line L(5, 372) { 30, 371 }: Central Line L(30, 371) { 3, 485 }: Central Line L(3, 485)
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_490 = CEVAPOINT OF X(20) AND X(488) (0.4037, -0.0241) 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_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7148, 0.4568) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5652, 0.2097) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_485 = VECTEN POINT (0.5773, 0.2578) X_3070 = INTERSECTION OF X(4)X(6) AND X(5)X(372) (0.6043, 0.3120)
A
B
C
Ma
Mb
Mc
X2
X490
X4
X6
X5
X372
X30
X371
X3
X485
X3070
(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, 490 }: Central Line L(2, 490) { 4, 6 }: Van Aubel Line { 5, 372 }: Central Line L(5, 372) { 30, 371 }: Central Line L(30, 371) { 3, 485 }: Central Line L(3, 485)
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_490 = CEVAPOINT OF X(20) AND X(488) (0.3600, -0.1557) 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_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7692, 0.4615) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5946, 0.1622) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_485 = VECTEN POINT (0.6057, 0.2400) X_3070 = INTERSECTION OF X(4)X(6) AND X(5)X(372) (0.6400, 0.3178)
A
B
C
Ma
Mb
Mc
X2
X490
X4
X6
X5
X372
X30
X371
X3
X485
X3070
(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, 490 }: Central Line L(2, 490) { 4, 6 }: Van Aubel Line { 5, 372 }: Central Line L(5, 372) { 30, 371 }: Central Line L(30, 371) { 3, 485 }: Central Line L(3, 485)
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_490 = CEVAPOINT OF X(20) AND X(488) (0.3820, -0.1589) 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_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7296, 0.4725) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5794, 0.1635) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_485 = VECTEN POINT (0.5889, 0.2429) X_3070 = INTERSECTION OF X(4)X(6) AND X(5)X(372) (0.6180, 0.3224)
A
B
C
Ma
Mb
Mc
X2
X490
X4
X6
X5
X372
X30
X371
X3
X485
X3070
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, 490 }: Central Line L(2, 490) { 4, 6 }: Van Aubel Line { 5, 372 }: Central Line L(5, 372) { 30, 371 }: Central Line L(30, 371) { 3, 485 }: Central Line L(3, 485)
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_490 = CEVAPOINT OF X(20) AND X(488) (0.3305, -0.3811) 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_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7352, 0.3832) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.6182, 0.0560) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_485 = VECTEN POINT (0.6212, 0.2100) X_3070 = INTERSECTION OF X(4)X(6) AND X(5)X(372) (0.6513, 0.3306)
A
B
C
Ma
Mb
Mc
X2
X490
X4
X6
X5
X372
X30
X371
X3
X485
X3070
Kimberling (6, 9, 13) | AB = 1
X3070 (INTERSECTION OF X4 X_6 AND X5 X_372 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sin A + 2 \cos B \cos C : :\)
Barycentrics: \((\sin A)(\sin A + 2 \cos B \cos C) : :\)
Search Value (6, 9, 13): \(-0.665589545679\)
X3070
▸ Incident Central Lines11 lines
{2 490 3 485 4 6 5 372 20 1151 30 371 487 1991 642 2482 1124 1479 1335 1478 1837 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 X3070 Randy Hutson , July 11, 2019)
