X3835
X693 com(3rd EULER TRIANGLE)
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, 649 }: Central Line L(2, 649) { 512, 625 }: Central Line L(512, 625) { 514, 661 }: Central Line L(514, 661) { 650, 812 }: Central Line L(650, 812) { 522, 1491 }: Central Line L(522, 1491)
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_649 = CROSSDIFFERENCE OF X(1) and X(2) (-21.0000, 14.6969) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_625 = MIDPOINT OF X(623) AND X(624) (0.2068, -0.3280) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_812 = ODD (- 1, 1) INFINITY POINT (0.1586, -0.2483) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0318, 0.3430) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (25.5000, -10.6145) X_3835 = com(3rd EULER TRIANGLE) (11.3061, -7.0485)
A
B
C
Ma
Mb
Mc
X2
X649
X512
X625
X514
X661
X650
X812
X522
X1491
X3835
(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, 649 }: Central Line L(2, 649) { 512, 625 }: Central Line L(512, 625) { 514, 661 }: Central Line L(514, 661) { 650, 812 }: Central Line L(650, 812) { 522, 1491 }: Central Line L(522, 1491)
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_649 = CROSSDIFFERENCE OF X(1) and X(2) (-8.0000, 6.0000) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) X_625 = MIDPOINT OF X(623) AND X(624) (0.3433, -0.1486) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0889, -0.0314) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (20.8000, -8.4000) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (6.4000, -1.2000) X_812 = ODD (- 1, 1) INFINITY POINT (0.1375, -0.2440) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0226, 0.2413) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (11.4000, -3.7000) X_3835 = com(3rd EULER TRIANGLE) (4.8200, -2.7600)
A
B
C
Ma
Mb
Mc
X2
X649
X512
X625
X514
X661
X650
X812
X522
X1491
X3835
(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, 649 }: Central Line L(2, 649) { 512, 625 }: Central Line L(512, 625) { 514, 661 }: Central Line L(514, 661) { 650, 812 }: Central Line L(650, 812) { 522, 1491 }: Central Line L(522, 1491)
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_649 = CROSSDIFFERENCE OF X(1) and X(2) (-9.6244, 5.9482) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) X_625 = MIDPOINT OF X(623) AND X(624) (0.3568, -0.1733) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0910, -0.0001) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (24.7569, -8.3524) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (7.5663, -1.2021) X_812 = ODD (- 1, 1) INFINITY POINT (0.0873, -0.1934) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0316, 0.2303) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (13.5145, -3.6762) X_3835 = com(3rd EULER TRIANGLE) (5.6212, -2.7312)
A
B
C
Ma
Mb
Mc
X2
X649
X512
X625
X514
X661
X650
X812
X522
X1491
X3835
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, 649 }: Central Line L(2, 649) { 512, 625 }: Central Line L(512, 625) { 514, 661 }: Central Line L(514, 661) { 650, 812 }: Central Line L(650, 812) { 522, 1491 }: Central Line L(522, 1491)
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_649 = CROSSDIFFERENCE OF X(1) and X(2) (-2.9286, 1.6903) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) X_625 = MIDPOINT OF X(623) AND X(624) (0.4494, -0.0504) X_514 = ISOGONAL CONJUGATE OF X(101) (1.1917, -0.0133) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (9.8407, -1.9504) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (3.4560, -0.1300) X_812 = ODD (- 1, 1) INFINITY POINT (-0.0449, -0.1950) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.1115, 0.1100) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (5.7851, -0.7941) X_3835 = com(3rd EULER TRIANGLE) (2.2809, -0.7051)
A
B
C
Ma
Mb
Mc
X2
X649
X512
X625
X514
X661
X650
X812
X522
X1491
X3835
Kimberling (6, 9, 13) | AB = 1
X3835 (X693 com(3rd EULER TRIANGLE) ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Barycentrics: \((b - c)(ab + ac - bc) : :\)
Search Value (6, 9, 13): \(-2.817180849095\)
▸ Incident Central Lines14 lines
{2 649 116 3259 226 3676 512 625 513 3716 514 661 522 1491 650 812 788 3741 802 3709 824 3004 1538 3309 1848 3064 2254 3667
▸ Functional Relations7 relations
Notes
Let A'B'C' be the Aquila triangle. Let A″ be the trilinear product B' *C' , and define B″ , C″ cyclically. A″B″C″ is homothetic to ABC at X3835 A'B'C' at X10 Randy Hutson , June 27, 2018)
