X877
INTERSECTION OF LINES X4 X_69 AND X99 X_112
Let
T1 be the
tangential triangle of hyperbola {
A ,
B ,
C ,
X99 ,PU(37)}}. Let
T2 be the tangential triangle of hyperbola {
A ,
B ,
C ,
X4 ,
X112 ,PU(39)}}. Let
A'B'C' be the vertex triangle of
T1 and
T2 . The lines
AA' ,
BB' ,
CC' concur in
X877 . (
Randy Hutson , December 29, 2015)
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) { 4, 69 }: Central Line L(4, 69) { 99, 112 }: Central Line L(99, 112)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_878 = ISOGONAL CONJUGATE OF X(877) (1.1192, 0.0431) X_4 = ORTHOCENTER (0.6122, 0.3958) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_99 = STEINER POINT (0.1109, 0.4322) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_877 = INTERSECTION OF LINES X(4)X(69) AND X(99)X(112) (0.8267, 0.7499)
A
B
C
X878
X4
X69
X99
X112
X877
(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) { 4, 69 }: Central Line L(4, 69) { 99, 112 }: Central Line L(99, 112) { 2, 3 }: Euler Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_878 = ISOGONAL CONJUGATE OF X(877) (1.2165, 0.0000) X_4 = ORTHOCENTER (0.6400, 0.4800) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_99 = STEINER POINT (0.1511, 0.3581) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6400, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_877 = INTERSECTION OF LINES X(4)X(69) AND X(99)X(112) (0.6400, 0.4800)
A
B
C
X878
X4
X69
X99
X112
X2
X3
X877
(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) { 4, 69 }: Central Line L(4, 69) { 99, 112 }: Central Line L(99, 112) { 2, 3 }: Euler Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_878 = ISOGONAL CONJUGATE OF X(877) (1.3090, 0.0000) X_4 = ORTHOCENTER (0.6180, 0.4859) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_99 = STEINER POINT (0.1910, 0.3931) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6180, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_877 = INTERSECTION OF LINES X(4)X(69) AND X(99)X(112) (0.6180, 0.4859)
A
B
C
X878
X4
X69
X99
X112
X2
X3
X877
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) { 4, 69 }: Central Line L(4, 69) { 99, 112 }: Central Line L(99, 112)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_878 = ISOGONAL CONJUGATE OF X(877) (1.9152, -0.2231) X_4 = ORTHOCENTER (0.6331, 0.8294) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_99 = STEINER POINT (0.2513, 0.2387) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6732, 0.2689) X_877 = INTERSECTION OF LINES X(4)X(69) AND X(99)X(112) (0.4265, 0.2513)
A
B
C
X878
X4
X69
X99
X112
X877
Kimberling (6, 9, 13) | AB = 1
X877 (INTERSECTION OF LINES X4 X_69 AND X99 X_112 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((a^{2}u^{2} - bcvw)/[a^{2}u(bv - cw)]\) , where
\(u : v : w\) =
X4 Search Value (6, 9, 13): \(1.927746931778\)
▸ Incident Central Lines2 lines
▸ Functional Relations3 relations
