X2964
TRILINEAR PRODUCT X61 *X62
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) { 1, 21 }: Central Line L(1, 21) { 484, 580 }: Central Line L(484, 580) { 36, 1399 }: Central Line L(36, 1399) { 109, 1393 }: Central Line L(109, 1393) { 162, 2166 }: Central Line L(162, 2166) { 163, 2179 }: Central Line L(163, 2179)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2216 = -ISOCONJUGATE OF X(570) (0.6037, 0.2921) X_1 = INCENTER (0.5714, 0.2333) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_484 = 1st EVANS PERSPECTOR (2.0952, 3.0327) X_580 = ORTHOGONAL OF X(284) (0.7089, 0.4470) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_1399 = -BETH CONJUGATE OF X(35) (0.6178, 0.2762) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_1393 = -BETH CONJUGATE OF X(5) (0.6196, 0.3615) X_162 = CEVAPOINT OF X(108) AND X(109) (0.4897, 0.6018) X_2166 = -ISOCONJUGATE OF X(50) (0.9476, -0.1665) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.3174, 0.5883) X_2179 = -ISOCONJUGATE OF X(95) (0.6272, 0.3514) X_2964 = TRILINEAR PRODUCT X(61)*X(62) (0.6488, 0.3350)
A
B
C
X2216
X1
X21
X484
X580
X36
X1399
X109
X1393
X162
X2166
X163
X2179
X2964
(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) { 1, 21 }: Central Line L(1, 21) { 484, 580 }: Central Line L(484, 580) { 36, 1399 }: Central Line L(36, 1399) { 109, 1393 }: Central Line L(109, 1393) { 162, 2166 }: Central Line L(162, 2166) { 163, 2179 }: Central Line L(163, 2179)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2216 = -ISOCONJUGATE OF X(570) (0.6394, 0.2777) X_1 = INCENTER (0.6000, 0.2000) X_21 = SCHIFFLER POINT (0.5368, 0.1263) X_484 = 1st EVANS PERSPECTOR (1.4000, 1.8000) X_580 = ORTHOGONAL OF X(284) (0.7692, 0.4615) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.0000, 1.0000) X_1399 = -BETH CONJUGATE OF X(35) (0.6667, 0.2381) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_1393 = -BETH CONJUGATE OF X(5) (0.6462, 0.3692) X_162 = CEVAPOINT OF X(108) AND X(109) (0.6400, 0.4800) X_2166 = -ISOCONJUGATE OF X(50) (0.9795, -0.3991) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.4407, 0.5085) X_2179 = -ISOCONJUGATE OF X(95) (0.6569, 0.3519) X_2964 = TRILINEAR PRODUCT X(61)*X(62) (0.7021, 0.3191)
A
B
C
X2216
X1
X21
X484
X580
X36
X1399
X109
X1393
X162
X2166
X163
X2179
X2964
(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) { 1, 21 }: Central Line L(1, 21) { 484, 580 }: Central Line L(484, 580) { 36, 1399 }: Central Line L(36, 1399) { 109, 1393 }: Central Line L(109, 1393) { 162, 2166 }: Central Line L(162, 2166) { 163, 2179 }: Central Line L(163, 2179)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2216 = -ISOCONJUGATE OF X(570) (0.6177, 0.2821) X_1 = INCENTER (0.5841, 0.2021) X_21 = SCHIFFLER POINT (0.5310, 0.1276) X_484 = 1st EVANS PERSPECTOR (1.2932, 1.9071) X_580 = ORTHOGONAL OF X(284) (0.7296, 0.4725) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.9387, 1.0546) X_1399 = -BETH CONJUGATE OF X(35) (0.6411, 0.2416) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_1393 = -BETH CONJUGATE OF X(5) (0.6233, 0.3743) X_162 = CEVAPOINT OF X(108) AND X(109) (0.6180, 0.4859) X_2166 = -ISOCONJUGATE OF X(50) (0.9584, -0.5344) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.4503, 0.5059) X_2179 = -ISOCONJUGATE OF X(95) (0.6326, 0.3570) X_2964 = TRILINEAR PRODUCT X(61)*X(62) (0.6717, 0.3251)
A
B
C
X2216
X1
X21
X484
X580
X36
X1399
X109
X1393
X162
X2166
X163
X2179
X2964
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) { 1, 21 }: Central Line L(1, 21) { 484, 580 }: Central Line L(484, 580) { 36, 1399 }: Central Line L(36, 1399) { 109, 1393 }: Central Line L(109, 1393) { 162, 2166 }: Central Line L(162, 2166) { 163, 2179 }: Central Line L(163, 2179)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2216 = -ISOCONJUGATE OF X(570) (0.6370, 0.2607) X_1 = INCENTER (0.6154, 0.1300) X_21 = SCHIFFLER POINT (0.5385, 0.0448) X_484 = 1st EVANS PERSPECTOR (0.8087, 0.8081) X_580 = ORTHOGONAL OF X(284) (0.7582, 0.4476) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.7120, 0.4690) X_1399 = -BETH CONJUGATE OF X(35) (0.7534, 0.0288) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_1393 = -BETH CONJUGATE OF X(5) (0.6274, 0.2813) X_162 = CEVAPOINT OF X(108) AND X(109) (0.7026, 0.2691) X_2166 = -ISOCONJUGATE OF X(50) (0.6300, 0.2939) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.5553, 0.2986) X_2179 = -ISOCONJUGATE OF X(95) (0.6211, 0.2838) X_2964 = TRILINEAR PRODUCT X(61)*X(62) (0.7318, 0.2591)
A
B
C
X2216
X1
X21
X484
X580
X36
X1399
X109
X1393
X162
X2166
X163
X2179
X2964
Kimberling (6, 9, 13) | AB = 1
X2964 (TRILINEAR PRODUCT X61 *X62 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sin(A + \pi /6) \sin(A - \pi /6) : :\)
Search Value (6, 9, 13): \(-0.561343491062\)
▸ Alternate Coordinates3 forms
Trilinears: \(a^{2} - R^{2} : :\) (C. Lozada, 9/07/2013)
Trilinears: \(\cos^{2} A - 3 \sin^{2} A : :\)
Trilinears: \(4 \sin^{2} A - 1 : :\)
▸ Incident Central Lines9 lines
{1 21 35 2361 36 1399 109 1393 162 2166 163 2179 484 580 2148 2180 2216 2962
▸ Functional Relations3 relations
Notes
Let OA OB OC be the Kosnita triangle . Let A″ be the trilinear product of the circumcircle intercepts of line OB OC . Define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X2964 Randy Hutson , July 31 2018)
