X1581
ISOGONAL CONJUGATE OF X1580
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) { 10, 257 }: Central Line L(10, 257) { 37, 256 }: Central Line L(37, 256) { 65, 291 }: Central Line L(65, 291) { 171, 292 }: Central Line L(171, 292) { 82, 662 }: Central Line L(82, 662) { 733, 831 }: Central Line L(733, 831)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1580 = POINT ALUDRA (1.3094, 1.2033) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5240, 0.1822) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_256 = 1st SHARYGIN POINT (0.5601, 0.2146) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_291 = 2nd SHARYGIN POINT (1.0249, 0.0364) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5819, 0.2525) X_292 = -HIRST INVERSE OF X(291) (1.0127, 0.0417) X_82 = ISOGONAL CONJUGATE OF X(38) (0.5943, 0.2703) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.1811, 0.4951) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0071, 0.0450) X_831 = o-(1, 3) CIRCUMCIRCLE POINT (0.1595, 0.4822) X_1581 = ISOGONAL CONJUGATE OF X(1580) (1.0304, 0.0330)
A
B
C
X1580
X10
X257
X37
X256
X65
X291
X171
X292
X82
X662
X733
X831
X1581
(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) { 10, 257 }: Central Line L(10, 257) { 37, 256 }: Central Line L(37, 256) { 65, 291 }: Central Line L(65, 291) { 171, 292 }: Central Line L(171, 292) { 82, 662 }: Central Line L(82, 662) { 733, 831 }: Central Line L(733, 831)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1580 = POINT ALUDRA (1.0580, 0.7344) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5285, 0.1384) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5787, 0.1787) X_256 = 1st SHARYGIN POINT (0.5863, 0.1764) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6400, 0.2800) X_291 = 2nd SHARYGIN POINT (1.0327, 0.0449) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.6121, 0.2242) X_292 = -HIRST INVERSE OF X(291) (1.0074, 0.0540) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6285, 0.2456) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.2597, 0.4299) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (0.9964, 0.0602) X_831 = o-(1, 3) CIRCUMCIRCLE POINT (0.2248, 0.4174) X_1581 = ISOGONAL CONJUGATE OF X(1580) (1.0435, 0.0384)
A
B
C
X1580
X10
X257
X37
X256
X65
X291
X171
X292
X82
X662
X733
X831
X1581
(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) { 10, 257 }: Central Line L(10, 257) { 37, 256 }: Central Line L(37, 256) { 65, 291 }: Central Line L(65, 291) { 171, 292 }: Central Line L(171, 292) { 82, 662 }: Central Line L(82, 662) { 733, 831 }: Central Line L(733, 831)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1580 = POINT ALUDRA (1.0000, 0.7862) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5239, 0.1399) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5661, 0.1805) X_256 = 1st SHARYGIN POINT (0.5723, 0.1779) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6180, 0.2838) X_291 = 2nd SHARYGIN POINT (1.0000, 0.0000) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5944, 0.2271) X_292 = -HIRST INVERSE OF X(291) (1.0000, 0.0000) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6084, 0.2491) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.2938, 0.4492) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0000, 0.0000) X_831 = o-(1, 3) CIRCUMCIRCLE POINT (0.2620, 0.4397) X_1581 = ISOGONAL CONJUGATE OF X(1580) (1.0000, 0.0000)
A
B
C
X1580
X10
X257
X37
X256
X65
X291
X171
X292
X82
X662
X733
X831
X1581
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) { 10, 257 }: Central Line L(10, 257) { 37, 256 }: Central Line L(37, 256) { 65, 291 }: Central Line L(65, 291) { 171, 292 }: Central Line L(171, 292) { 82, 662 }: Central Line L(82, 662) { 733, 831 }: Central Line L(733, 831)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1580 = POINT ALUDRA (0.8179, 0.3546) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5274, 0.0726) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5913, 0.1097) X_256 = 1st SHARYGIN POINT (0.6068, 0.1059) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6417, 0.2223) X_291 = 2nd SHARYGIN POINT (1.0107, 0.0104) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.6224, 0.1547) X_292 = -HIRST INVERSE OF X(291) (0.9966, 0.0147) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6382, 0.1738) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.3982, 0.2796) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (0.9896, 0.0181) X_831 = o-(1, 3) CIRCUMCIRCLE POINT (0.3508, 0.2759) X_1581 = ISOGONAL CONJUGATE OF X(1580) (1.0158, 0.0075)
A
B
C
X1580
X10
X257
X37
X256
X65
X291
X171
X292
X82
X662
X733
X831
X1581
Kimberling (6, 9, 13) | AB = 1
X1581 (ISOGONAL CONJUGATE OF X1580 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/(a^{4} - b^{2}c^{2}) : :\)
Search Value (6, 9, 13): \(-0.202161974065\)
▸ Incident Central Lines8 lines
{10 257 37 256 65 291 82 662 171 292 733 831 759 805 876 882
▸ Functional Relations9 relations
X1581 X1580
X1581 isotomic conjugate of X1966
X1581 cevapoint of Xi and Xj for these (i,j): (1755 1964 1965 1966
X1581 i -cross conjugate of Xj for these (i,j): (1821 1956 1959 1 2227 75
X1581 trilinear pole of PU(35) (line X38 X661
X1581
X1581 X256 X291
X1581
X1581 cevian triangles of PU(8)
Notes
Let DEF and D'E'F' be the 1st and 2nd Sharygin triangles. Let A' be the trilinear product D*D' , and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X1581 Randy Hutson , December 26, 2015)
