X1917
TRILINEAR 6th POWER POINT
X1917 = barycentric product of PU(12)
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) { 31, 2085 }: Central Line L(31, 2085) { 1, 3409 }: Central Line L(1, 3409)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1928 = ISOGONAL CONJUGATE OF X(1917) (0.3248, 0.0702) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_2085 = NK-TRANSFORM OF X(31) (0.5956, 0.2227) X_1 = INCENTER (0.5714, 0.2333) X_3409 = COT(A + 2ω) POINT (0.6532, 0.3973) X_1917 = TRILINEAR 6th POWER POINT (0.6636, 0.4181)
A
B
C
X1928
X31
X2085
X1
X3409
X1917
(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) { 31, 2085 }: Central Line L(31, 2085) { 1, 3409 }: Central Line L(1, 3409)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1928 = ISOGONAL CONJUGATE OF X(1917) (0.2183, 0.0284) X_31 = 2nd POWER POINT (0.6667, 0.2778) X_2085 = NK-TRANSFORM OF X(31) (0.6494, 0.1820) X_1 = INCENTER (0.6000, 0.2000) X_3409 = COT(A + 2ω) POINT (0.7003, 0.4176) X_1917 = TRILINEAR 6th POWER POINT (0.6865, 0.3878)
A
B
C
X1928
X31
X2085
X1
X3409
X1917
(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) { 31, 2085 }: Central Line L(31, 2085) { 1, 3409 }: Central Line L(1, 3409)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1928 = ISOGONAL CONJUGATE OF X(1917) (0.2560, 0.0315) X_31 = 2nd POWER POINT (0.6411, 0.2822) X_2085 = NK-TRANSFORM OF X(31) (0.6259, 0.1818) X_1 = INCENTER (0.5841, 0.2021) X_3409 = COT(A + 2ω) POINT (0.6679, 0.4224) X_1917 = TRILINEAR 6th POWER POINT (0.6587, 0.3982)
A
B
C
X1928
X31
X2085
X1
X3409
X1917
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) { 31, 2085 }: Central Line L(31, 2085) { 1, 3409 }: Central Line L(1, 3409)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1928 = ISOGONAL CONJUGATE OF X(1917) (0.1258, 0.0051) X_31 = 2nd POWER POINT (0.6747, 0.1958) X_2085 = NK-TRANSFORM OF X(31) (0.7014, 0.1037) X_1 = INCENTER (0.6154, 0.1300) X_3409 = COT(A + 2ω) POINT (0.6868, 0.3547) X_1917 = TRILINEAR 6th POWER POINT (0.6564, 0.2591)
A
B
C
X1928
X31
X2085
X1
X3409
X1917
Kimberling (6, 9, 13) | AB = 1
X1917 (TRILINEAR 6th POWER POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a^{6} : :\)
Search Value (6, 9, 13): \(0.032563318712\)
▸ Incident Central Lines5 lines
{1 3409 31 2085 560 922 1501
▸ Functional Relations5 relations
Notes
Let A'B'C' and A″B″C″ be the 5th Brocard and 5th anti-Brocard triangles, resp. Let A* be the trilinear product A' *A″ , and define B* and C* cyclically. The lines AA* , BB* , CC* concur in X1917 Randy Hutson , November 30, 2018)
