X563
TRILINEAR PRODUCT X47 *X48
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) { 19, 163 }: Central Line L(19, 163) { 48, 255 }: Central Line L(48, 255) { 1, 1820 }: Central Line L(1, 1820)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_19 = CLAWSON POINT (0.6246, 0.4244) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.3174, 0.5883) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4217, 0.0365) X_1 = INCENTER (0.5714, 0.2333) X_1820 = ISOGONAL CONJUGATE OF MIMOSA TRANSFORM OF X(264) (-0.0663, -0.0078) X_563 = TRILINEAR PRODUCT X(47)*X(48) (0.8122, 0.3243)
A
B
C
X19
X163
X48
X255
X1
X1820
X563
(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) { 19, 163 }: Central Line L(19, 163) { 48, 255 }: Central Line L(48, 255) { 1, 1820 }: Central Line L(1, 1820)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_19 = CLAWSON POINT (0.6400, 0.4800) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.4407, 0.5085) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5714, 0.0000) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4286, 0.0000) X_1 = INCENTER (0.6000, 0.2000) X_1820 = ISOGONAL CONJUGATE OF MIMOSA TRANSFORM OF X(264) (4.0000, 0.0000) X_563 = TRILINEAR PRODUCT X(47)*X(48) (4.0000, 0.0000)
A
B
C
X19
X163
X48
X255
X1
X1820
X563
(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) { 19, 163 }: Central Line L(19, 163) { 48, 255 }: Central Line L(48, 255) { 1, 1820 }: Central Line L(1, 1820)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_19 = CLAWSON POINT (0.6180, 0.4859) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.4503, 0.5059) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5599, 0.0000) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4401, 0.0000) X_1 = INCENTER (0.5841, 0.2021) X_1820 = ISOGONAL CONJUGATE OF MIMOSA TRANSFORM OF X(264) (4.6762, 0.0000) X_563 = TRILINEAR PRODUCT X(47)*X(48) (4.6762, 0.0000)
A
B
C
X19
X163
X48
X255
X1
X1820
X563
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) { 19, 163 }: Central Line L(19, 163) { 48, 255 }: Central Line L(48, 255) { 1, 1820 }: Central Line L(1, 1820)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_19 = CLAWSON POINT (0.5737, 0.4687) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.5553, 0.2986) X_48 = CROSSPOINT OF X(1) AND X(63) (0.8073, -1.4298) X_255 = ISOGONAL CONJUGATE OF X(158) (0.5536, 0.0615) X_1 = INCENTER (0.6154, 0.1300) X_1820 = ISOGONAL CONJUGATE OF MIMOSA TRANSFORM OF X(264) (0.7649, 0.0955) X_563 = TRILINEAR PRODUCT X(47)*X(48) (0.5389, 0.1477)
A
B
C
X19
X163
X48
X255
X1
X1820
X563
Kimberling (6, 9, 13) | AB = 1
X563 (TRILINEAR PRODUCT X47 *X48 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sin 4A : \sin 4B : \sin 4C\)
Barycentrics: \(\sin A \sin 4A : :\)
Search Value (6, 9, 13): \(2.110873742995\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines3 lines
{1 1820 19 163 48 255
▸ Functional Relations1 relation
Notes
Clark Kimberling and Cyril Parry, “Products, Square Roots, and Layers in Triangle Geometry,” Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
