X153
REFLECTION OF X20 IN X100
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) { 20, 100 }: Central Line L(20, 100) { 7, 80 }: Central Line L(7, 80) { 2, 104 }: Central Line L(2, 104) { 4, 145 }: Central Line L(4, 145) { 11, 388 }: Central Line L(11, 388)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_7 = GERGONNE POINT (0.5903, 0.2769) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_2 = CENTROID (0.5374, 0.2000) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_4 = ORTHOCENTER (0.6122, 0.3958) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_388 = ∩X(7)X(8) (0.5790, 0.2635) X_153 = REFLECTION OF X(20) IN X(100) (-0.1020, 1.1248)
A
B
C
X20
X100
X7
X80
X2
X104
X4
X145
X11
X388
X153
(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) { 20, 100 }: Central Line L(20, 100) { 7, 80 }: Central Line L(7, 80) { 2, 104 }: Central Line L(2, 104) { 4, 145 }: Central Line L(4, 145) { 11, 388 }: Central Line L(11, 388)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_20 = DE LONGCHAMPS POINT (0.3600, -0.4800) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_7 = GERGONNE POINT (0.6218, 0.2618) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8400, -0.1200) X_2 = CENTROID (0.5467, 0.1600) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_4 = ORTHOCENTER (0.6400, 0.4800) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.7600, 0.3200) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_388 = ∩X(7)X(8) (0.6067, 0.2467) X_153 = REFLECTION OF X(20) IN X(100) (0.0400, 1.2800)
A
B
C
X20
X100
X7
X80
X2
X104
X4
X145
X11
X388
X153
(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) { 20, 100 }: Central Line L(20, 100) { 7, 80 }: Central Line L(7, 80) { 2, 104 }: Central Line L(2, 104) { 4, 145 }: Central Line L(4, 145) { 11, 388 }: Central Line L(11, 388)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_20 = DE LONGCHAMPS POINT (0.3820, -0.4859) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_7 = GERGONNE POINT (0.6026, 0.2652) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.7953, -0.1425) X_2 = CENTROID (0.5393, 0.1620) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_4 = ORTHOCENTER (0.6180, 0.4859) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.7182, 0.3225) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_388 = ∩X(7)X(8) (0.5898, 0.2498) X_153 = REFLECTION OF X(20) IN X(100) (0.0953, 1.3384)
A
B
C
X20
X100
X7
X80
X2
X104
X4
X145
X11
X388
X153
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) { 20, 100 }: Central Line L(20, 100) { 7, 80 }: Central Line L(7, 80) { 2, 104 }: Central Line L(2, 104) { 4, 145 }: Central Line L(4, 145) { 11, 388 }: Central Line L(11, 388)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_20 = DE LONGCHAMPS POINT (0.3669, -1.3788) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_7 = GERGONNE POINT (0.6288, 0.2114) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.6972, -0.1168) X_2 = CENTROID (0.5444, 0.0934) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_4 = ORTHOCENTER (0.6331, 0.8294) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.8284, 0.2400) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_388 = ∩X(7)X(8) (0.6172, 0.2016) X_153 = REFLECTION OF X(20) IN X(100) (0.2743, 1.9125)
A
B
C
X20
X100
X7
X80
X2
X104
X4
X145
X11
X388
X153
Kimberling (6, 9, 13) | AB = 1
X153 (REFLECTION OF X20 IN X100 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where
\(f(a,b,c) = (by + cz - ax)/a\) , where
\(x : y : z\) =
X104 Barycentrics: \(a^{7} - a^{6} (b + c) - a^{5} (b^{2} - 7 b c + c^{2}) + a^{4} (b + c) (b^{2} - 6 b c + c^{2}) - a^{3} (b^{4} + 2 b^{3} c - 10 b^{2} c^{2} + 2 b c^{3} + c^{4}) + a^{2} (b - c)^{2} (b + c) (b^{2} + 6 b c + c^{2}) + a (b^{2} - c^{2})^{2} (b^{2} - 5 b c + c^{2}) - (b - c)^{4} (b + c)^{3} : :\)
Search Value (6, 9, 13): \(-14.037672929368\)
X153
▸ Incident Central Lines6 lines
{2 104 4 145 7 80 11 388 20 100 515 908
▸ Functional Relations4 relations
Notes
Let T be the P -Brocard triangle for any P on line X1 X3 OA be the circle centered at the A -vertex of T and passing through A ; define OB and OC cyclically. X153 OA , OB , OC . (Randy Hutson , August 30, 2020)
