X3346
ISOGONAL CONJUGATE OF X1498
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) { 3183, 3 }: Central Line L(3183, 3) { 20, 394 }: Central Line L(20, 394) { 3, 1033 }: Central Line L(3, 1033) { 4, 1073 }: Central Line L(4, 1073) { 40, 3182 }: Central Line L(40, 3182)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3183 = -CEVA CONJUGATE OF X(4) (0.5835, 1.0296) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_394 = -CEVA CONJUGATE OF X(3) (0.3778, 0.0284) X_1033 = -CEVA CONJUGATE OF X(25) (0.5778, 0.9664) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1073 = -CROSS CONJUGATE OF X(3) (0.4792, -0.4343) X_40 = BEVAN POINT (0.4286, -0.0292) X_3182 = -CEVA CONJUGATE OF X(1) (0.5164, 5.7648) X_3346 = ISOGONAL CONJUGATE OF X(1498) (0.4165, -0.8255)
A
B
C
X3183
X3
X20
X394
X1033
X4
X1073
X40
X3182
X3346
(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) { 3183, 3 }: Euler Line { 2, 3 }: Euler Line { 20, 394 }: Central Line L(20, 394) { 3, 1033 }: Central Line L(3, 1033) { 4, 1073 }: Euler Line { 40, 3182 }: Central Line L(40, 3182)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3183 = -CEVA CONJUGATE OF X(4) (0.6400, 0.4800) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_2 = CENTROID (0.5467, 0.1600) X_20 = DE LONGCHAMPS POINT (0.3600, -0.4800) X_394 = -CEVA CONJUGATE OF X(3) (0.3600, 0.0000) X_1033 = -CEVA CONJUGATE OF X(25) (0.5000, 0.5208) X_4 = ORTHOCENTER (0.6400, 0.4800) X_1073 = -CROSS CONJUGATE OF X(3) (0.5000, 0.0000) X_40 = BEVAN POINT (0.4000, -0.2000) X_3182 = -CEVA CONJUGATE OF X(1) (0.6000, 1.2000) X_3346 = ISOGONAL CONJUGATE OF X(1498) (0.3600, -0.4800)
A
B
C
X3183
X3
X2
X20
X394
X1033
X4
X1073
X40
X3182
X3346
(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) { 3183, 3 }: Euler Line { 2, 3 }: Euler Line { 20, 394 }: Central Line L(20, 394) { 3, 1033 }: Central Line L(3, 1033) { 4, 1073 }: Euler Line { 40, 3182 }: Central Line L(40, 3182)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3183 = -CEVA CONJUGATE OF X(4) (0.6180, 0.4859) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_2 = CENTROID (0.5393, 0.1620) X_20 = DE LONGCHAMPS POINT (0.3820, -0.4859) X_394 = -CEVA CONJUGATE OF X(3) (0.3820, 0.0000) X_1033 = -CEVA CONJUGATE OF X(25) (0.5000, 0.5145) X_4 = ORTHOCENTER (0.6180, 0.4859) X_1073 = -CROSS CONJUGATE OF X(3) (0.5000, 0.0000) X_40 = BEVAN POINT (0.4159, -0.2021) X_3182 = -CEVA CONJUGATE OF X(1) (0.5841, 1.2021) X_3346 = ISOGONAL CONJUGATE OF X(1498) (0.3820, -0.4859)
A
B
C
X3183
X3
X2
X20
X394
X1033
X4
X1073
X40
X3182
X3346
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) { 3183, 3 }: Central Line L(3183, 3) { 20, 394 }: Central Line L(20, 394) { 3, 1033 }: Central Line L(3, 1033) { 4, 1073 }: Central Line L(4, 1073) { 40, 3182 }: Central Line L(40, 3182)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3183 = -CEVA CONJUGATE OF X(4) (0.5771, -0.0414) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_20 = DE LONGCHAMPS POINT (0.3669, -1.3788) X_394 = -CEVA CONJUGATE OF X(3) (0.4580, 0.0382) X_1033 = -CEVA CONJUGATE OF X(25) (0.6616, 0.2142) X_4 = ORTHOCENTER (0.6331, 0.8294) X_1073 = -CROSS CONJUGATE OF X(3) (0.5128, 0.0639) X_40 = BEVAN POINT (0.3846, -0.6794) X_3182 = -CEVA CONJUGATE OF X(1) (0.6404, 0.4669) X_3346 = ISOGONAL CONJUGATE OF X(1498) (0.4229, -0.5079)
A
B
C
X3183
X3
X20
X394
X1033
X4
X1073
X40
X3182
X3346
Kimberling (6, 9, 13) | AB = 1
X3346 (ISOGONAL CONJUGATE OF X1498 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[(\sin A)(\tan^{2} A - \tan^{2} B - \tan^{2} C)] : :\)
Search Value (6, 9, 13): \(9.801004737818\)
X3346
▸ Incident Central Lines7 lines
{1 3353 3 1033 4 1073 20 394 40 3182 64 3355 84 3472
▸ Functional Relations9 relations
Notes
Let A'B'C' be the cevian triangle of X20 A″ be the orthocenter of AB'C' , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X3346 Randy Hutson , November 18, 2015)
Let M be a point on the slideline BC of triangle ABC . Let (AM) be the circle with diameter AM. Let Mb be the point, other than A, in (AM)∩AC , and let Mc be the point, other than A, in (AM)∩AB . Let Tb be the line tangent to (AM) at Mb , and let Tc be the line tangent to (AM) at Mc . As M varies on BC , the locus of Tb ∩Tc is a line, La . Define Lb and Lc cyclically. Let A' = Lb ∩Lc , and define B' and C' cyclically. The triangle A'B'C' is perspective to ABCF, and the perspector is X3346 Angel Montesdeoca , November 11, 2017)
