X3142
X109 -CEVA CONJUGATE OF X523
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) { 2, 3 }: Euler Line { 12, 73 }: Central Line L(12, 73) { 117, 125 }: Central Line L(117, 125) { 11, 1193 }: Central Line L(11, 1193) { 226, 1425 }: Central Line L(226, 1425) { 1211, 1329 }: Central Line L(1211, 1329)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_117 = MIDPOINT OF X(4) AND X(109) (0.4879, 0.4948) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_1193 = 3rd SARAGOSSA POINT OF X(1) (0.5959, 0.2573) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_1425 = -BETH CONJUGATE OF X(201) (0.5904, 0.1922) X_1211 = ISOGONAL CONJUGATE OF X(1169) (0.5137, 0.1744) X_1329 = COMPLEMENTARY CONJUGATE OF X(1) (0.4903, 0.1280) X_3142 = -CEVA CONJUGATE OF X(523) (0.5719, 0.2902)
A
B
C
X109
X523
X2
X3
X12
X73
X117
X125
X11
X1193
X226
X1425
X1211
X1329
X3142
(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) { 2, 3 }: Euler Line { 12, 73 }: Central Line L(12, 73) { 117, 125 }: Central Line L(117, 125) { 11, 1193 }: Central Line L(11, 1193) { 226, 1425 }: Central Line L(226, 1425) { 1211, 1329 }: Central Line L(1211, 1329)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0987, -0.0010) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5714, 0.0000) X_117 = MIDPOINT OF X(4) AND X(109) (0.5700, 0.4900) X_125 = CENTER OF JERABEK HYPERBOLA (0.6400, 0.0000) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_1193 = 3rd SARAGOSSA POINT OF X(1) (0.6364, 0.2273) X_226 = -CEVA CONJUGATE OF X(65) (0.6057, 0.2400) X_1425 = -BETH CONJUGATE OF X(201) (0.6400, 0.0000) X_1211 = ISOGONAL CONJUGATE OF X(1169) (0.5108, 0.1292) X_1329 = COMPLEMENTARY CONJUGATE OF X(1) (0.4867, 0.0733) X_3142 = -CEVA CONJUGATE OF X(523) (0.5940, 0.3222)
A
B
C
X109
X523
X2
X3
X12
X73
X117
X125
X11
X1193
X226
X1425
X1211
X1329
X3142
(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) { 2, 3 }: Euler Line { 12, 73 }: Central Line L(12, 73) { 117, 125 }: Central Line L(117, 125) { 11, 1193 }: Central Line L(11, 1193) { 226, 1425 }: Central Line L(226, 1425) { 1211, 1329 }: Central Line L(1211, 1329)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0981, 0.0262) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5599, 0.0000) X_117 = MIDPOINT OF X(4) AND X(109) (0.5590, 0.4929) X_125 = CENTER OF JERABEK HYPERBOLA (0.6180, 0.0000) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_1193 = 3rd SARAGOSSA POINT OF X(1) (0.6149, 0.2298) X_226 = -CEVA CONJUGATE OF X(65) (0.5889, 0.2429) X_1425 = -BETH CONJUGATE OF X(201) (0.6180, 0.0000) X_1211 = ISOGONAL CONJUGATE OF X(1169) (0.5092, 0.1309) X_1329 = COMPLEMENTARY CONJUGATE OF X(1) (0.4885, 0.0733) X_3142 = -CEVA CONJUGATE OF X(523) (0.5791, 0.3255)
A
B
C
X109
X523
X2
X3
X12
X73
X117
X125
X11
X1193
X226
X1425
X1211
X1329
X3142
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) { 2, 3 }: Euler Line { 12, 73 }: Central Line L(12, 73) { 117, 125 }: Central Line L(117, 125) { 11, 1193 }: Central Line L(11, 1193) { 226, 1425 }: Central Line L(226, 1425) { 1211, 1329 }: Central Line L(1211, 1329)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_523 = ISOGONAL CONJUGATE OF X(110) (1.1957, 0.0148) X_2 = CENTROID (0.5444, 0.0934) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.6250, 0.5094) X_117 = MIDPOINT OF X(4) AND X(109) (0.6190, 0.5577) X_125 = CENTER OF JERABEK HYPERBOLA (0.5678, -0.0079) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_1193 = 3rd SARAGOSSA POINT OF X(1) (0.6586, 0.1523) X_226 = -CEVA CONJUGATE OF X(65) (0.6170, 0.1949) X_1425 = -BETH CONJUGATE OF X(201) (0.6239, 0.3121) X_1211 = ISOGONAL CONJUGATE OF X(1169) (0.4997, 0.0648) X_1329 = COMPLEMENTARY CONJUGATE OF X(1) (0.4918, 0.0153) X_3142 = -CEVA CONJUGATE OF X(523) (0.6747, 1.1742)
A
B
C
X109
X523
X2
X3
X12
X73
X117
X125
X11
X1193
X226
X1425
X1211
X1329
X3142
Kimberling (6, 9, 13) | AB = 1
X3142 (X109 -CEVA CONJUGATE OF X523 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\) , where
\(x : y : z\) =
X109 and
\(u : v : w\) =
X523 Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\) , where \(f(a,b,c) = u(-u/x + v/y + w/z)\)
Search Value (6, 9, 13): \(-9.567248245795\)
▸ Incident Central Lines9 lines
{2 3 11 1193 12 73 117 125 226 1425 1211 1329 1213 2183 1214 1867 1400 1901
▸ Functional Relations3 relations
Notes
As a point on the Euler line, X3142 \((\$aS_{A}^{2}S_{B}\$ -\$aS_{C}S_{A}^{2}\$-\$aS_{A}S_{B}\$F +\$aS_{C}S_{A}\$F, \$aS_{B}^{2}S_{C}\$ -\$aS_{B}S_{C}^{2}\$-\$aS_{B}\$S^{2} +\$aS_{C}\$S^{2})\) .
