X1173
1st SARAGOSSA POINT OF X54
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) { 51, 54 }: Central Line L(51, 54) { 3, 143 }: Central Line L(3, 143) { 74, 389 }: Central Line L(74, 389) { 69, 576 }: Central Line L(69, 576) { 265, 546 }: Central Line L(265, 546) { 575, 1176 }: Central Line L(575, 1176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_54 = KOSNITA POINT (0.5880, 0.2176) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6078, 0.2800) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_576 = ORTHOHARMONIC OF X(187) (0.6500, 0.3498) X_265 = REFLECTION OF X(3) IN X(125) (0.8606, -0.0500) X_546 = MIDPOINT OF X(4) AND X(5) (0.5842, 0.3223) X_575 = ORTHOGONAL OF X(187) (0.5750, 0.2259) X_1176 = 1st SARAGOSSA POINT OF X(66) (0.5328, 0.1263) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6043, 0.2952)
A
B
C
X140
X51
X54
X3
X143
X74
X389
X69
X576
X265
X546
X575
X1176
X1173
(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) { 51, 54 }: Van Aubel Line { 3, 143 }: Central Line L(3, 143) { 74, 389 }: Van Aubel Line { 69, 576 }: Central Line L(69, 576) { 265, 546 }: Central Line L(265, 546) { 575, 1176 }: Central Line L(575, 1176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_140 = MIDPOINT OF X(3) AND X(5) (0.5350, 0.1200) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6400, 0.3200) X_54 = KOSNITA POINT (0.6400, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.7120, 0.2331) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6400, -0.4800) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6400, 0.2400) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_576 = ORTHOHARMONIC OF X(187) (0.7100, 0.3600) X_265 = REFLECTION OF X(3) IN X(125) (0.7800, 0.0000) X_546 = MIDPOINT OF X(4) AND X(5) (0.6050, 0.3600) X_575 = ORTHOGONAL OF X(187) (0.6050, 0.1800) X_1176 = 1st SARAGOSSA POINT OF X(66) (0.5467, 0.0000) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6400, 0.2880)
A
B
C
X140
X51
X54
X3
X143
X74
X389
X69
X576
X265
X546
X575
X1176
X1173
(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) { 51, 54 }: Van Aubel Line { 3, 143 }: Central Line L(3, 143) { 74, 389 }: Van Aubel Line { 69, 576 }: Central Line L(69, 576) { 265, 546 }: Central Line L(265, 546) { 575, 1176 }: Central Line L(575, 1176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_140 = MIDPOINT OF X(3) AND X(5) (0.5295, 0.1215) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6180, 0.3239) X_54 = KOSNITA POINT (0.6180, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6491, 0.3580) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6180, -0.4859) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6180, 0.2429) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_576 = ORTHOHARMONIC OF X(187) (0.6771, 0.3644) X_265 = REFLECTION OF X(3) IN X(125) (0.7361, 0.0000) X_546 = MIDPOINT OF X(4) AND X(5) (0.5885, 0.3644) X_575 = ORTHOGONAL OF X(187) (0.5885, 0.1822) X_1176 = 1st SARAGOSSA POINT OF X(66) (0.5393, 0.0000) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6180, 0.2915)
A
B
C
X140
X51
X54
X3
X143
X74
X389
X69
X576
X265
X546
X575
X1176
X1173
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) { 51, 54 }: Central Line L(51, 54) { 3, 143 }: Central Line L(3, 143) { 74, 389 }: Central Line L(74, 389) { 69, 576 }: Central Line L(69, 576) { 265, 546 }: Central Line L(265, 546) { 575, 1176 }: Central Line L(575, 1176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_140 = MIDPOINT OF X(3) AND X(5) (0.5333, 0.0013) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6126, 0.2841) X_54 = KOSNITA POINT (1.8800, -0.0730) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6356, 0.2874) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.5025, -0.8451) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6023, 0.0114) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_576 = ORTHOHARMONIC OF X(187) (0.7360, 0.3856) X_265 = REFLECTION OF X(3) IN X(125) (0.6357, 0.2589) X_546 = MIDPOINT OF X(4) AND X(5) (0.5999, 0.5534) X_575 = ORTHOGONAL OF X(187) (0.6180, 0.0554) X_1176 = 1st SARAGOSSA POINT OF X(66) (0.3063, -4.4755) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6333, 0.2782)
A
B
C
X140
X51
X54
X3
X143
X74
X389
X69
X576
X265
X546
X575
X1176
X1173
Kimberling (6, 9, 13) | AB = 1
X1173 (1st SARAGOSSA POINT OF X54 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a/[2a^{4} - 3a^{2}(b^{2} + c^{2}) + (b^{2} - c^{2})^{2}] : :\)
Search Value (6, 9, 13): \(0.017230908098\)
▸ Incident Central Lines6 lines
{3 143 51 54 69 576 74 389 265 546 575 1176
▸ Functional Relations6 relations
Notes
Let P and Q be the intersections of line BC and the 2nd Lemoine circle. Let X = X6 OA be the circumcenter of triangle PQX , and define OB and OC cyclically. The lines AOA , BOB , COC concur in X1173 X592 X571 OA OB OC be the Kosnita triangle . Let A″ be the barycentric product OB *OC , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X571 Randy Hutson , December 2, 2017)
Let Ha be the foot of the A -altitude. Let Ba , Ca be the feet of perpendiculars from Ha to CA , AB , resp. Let Na be the nine-point center of Ha Ba Ca . Define Nb and Nc cyclically. The lines ANa , BNb , CNc concur in X1173
