X3000
(ANTIORTHIC AXIS)∩(SODDY LINE)
The antiorthic axis is line
X44 X513 , more simply represented as L(1), meaning the line whose coefficients in trilinear coordinates are
\(1 : 1 : 1\) , these also being trilinears for
X1 . The Soddy line is
X1 X7 , alias L(657). For such identifications as these, use the MathWorld link just before
X3000 .
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) { 2310, 241 }: Central Line L(2310, 241) { 1, 7 }: Soddy Line { 44, 513 }: Central Line L(44, 513) { 527, 2340 }: Central Line L(527, 2340)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2310 = -ISOCONJUGATE OF X(1262) (0.7196, 0.0518) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8698, 0.6836) X_2340 = -ISOCONJUGATE OF X(1462) (-1.1481, -1.4515) X_3000 = (ANTIORTHIC AXIS)∩(SODDY LINE) (1.5238, 2.4365)
A
B
C
X2310
X241
X1
X7
X44
X513
X527
X2340
X3000
(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) { 2310, 241 }: Central Line L(2310, 241) { 1, 7 }: Soddy Line { 44, 513 }: Central Line L(44, 513) { 527, 2340 }: Central Line L(527, 2340)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2310 = -ISOCONJUGATE OF X(1262) (0.7000, 0.0250) X_241 = -LINE CONJUGATE OF X(55) (0.8750, 0.7500) X_1 = INCENTER (0.6000, 0.2000) X_7 = GERGONNE POINT (0.6218, 0.2618) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8881, 0.6226) X_2340 = -ISOCONJUGATE OF X(1462) (-0.5000, -0.6250) X_3000 = (ANTIORTHIC AXIS)∩(SODDY LINE) (1.0500, 1.4750)
A
B
C
X2310
X241
X1
X7
X44
X513
X527
X2340
X3000
(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) { 2310, 241 }: Central Line L(2310, 241) { 1, 7 }: Soddy Line { 44, 513 }: Central Line L(44, 513) { 527, 2340 }: Central Line L(527, 2340)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2310 = -ISOCONJUGATE OF X(1262) (0.6709, 0.0183) X_241 = -LINE CONJUGATE OF X(55) (0.8238, 0.7785) X_1 = INCENTER (0.5841, 0.2021) X_7 = GERGONNE POINT (0.6026, 0.2652) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8397, 0.6523) X_2340 = -ISOCONJUGATE OF X(1462) (-0.3626, -0.6477) X_3000 = (ANTIORTHIC AXIS)∩(SODDY LINE) (0.9767, 1.5388)
A
B
C
X2310
X241
X1
X7
X44
X513
X527
X2340
X3000
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) { 2310, 241 }: Central Line L(2310, 241) { 1, 7 }: Soddy Line { 44, 513 }: Central Line L(44, 513) { 527, 2340 }: Central Line L(527, 2340)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2310 = -ISOCONJUGATE OF X(1262) (0.6421, 0.0019) X_241 = -LINE CONJUGATE OF X(55) (0.6720, 0.3286) X_1 = INCENTER (0.6154, 0.1300) X_7 = GERGONNE POINT (0.6288, 0.2114) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.9260, 0.6270) X_2340 = -ISOCONJUGATE OF X(1462) (0.1569, -0.1068) X_3000 = (ANTIORTHIC AXIS)∩(SODDY LINE) (0.7019, 0.6552)
A
B
C
X2310
X241
X1
X7
X44
X513
X527
X2340
X3000
Kimberling (6, 9, 13) | AB = 1
X3000 ((ANTIORTHIC AXIS)∩(SODDY LINE) ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where \(f(a,b,c) = (b + c)a^{3} + 2(bc - b^{2} - c^{2})a^{2} + (ab + ac - 2bc)(b - c)^{2}\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-4.419091875470\)
▸ Incident Central Lines4 lines
{1 7 44 513 241 2310 527 2340
▸ Functional Relations2 relations
