X827
o-(0, 4) CIRCUMCIRCLE POINT
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) { 5, 83 }: Central Line L(5, 83) { 111, 251 }: Central Line L(111, 251) { 6, 755 }: Central Line L(6, 755) { 32, 733 }: Central Line L(32, 733) { 31, 745 }: Central Line L(31, 745) { 82, 759 }: Central Line L(82, 759)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_826 = ODD (0, 4) INFINITY POINT (1.1003, 0.0340) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_83 = CEVAPOINT OF CENTROID AND SYMMEDIAN POINT (0.5651, 0.2360) X_111 = PARRY POINT (0.9885, -0.0457) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6177, 0.3042) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_755 = EVEN (0, 4) CIRCUMCIRCLE POINT (0.9524, -0.1340) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0071, 0.0450) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_745 = EVEN (- 1, 3) CIRCUMCIRCLE POINT (0.9892, -0.0432) X_82 = ISOGONAL CONJUGATE OF X(38) (0.5943, 0.2703) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (0.9498, -0.1389) X_827 = o-(0, 4) CIRCUMCIRCLE POINT (0.3242, 0.5811)
A
B
C
X826
X5
X83
X111
X251
X6
X755
X32
X733
X31
X745
X82
X759
X827
(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) { 5, 83 }: Central Line L(5, 83) { 111, 251 }: Central Line L(111, 251) { 6, 755 }: Central Line L(6, 755) { 32, 733 }: Central Line L(32, 733) { 31, 745 }: Central Line L(31, 745) { 82, 759 }: Central Line L(82, 759)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_826 = ODD (0, 4) INFINITY POINT (1.1111, 0.0502) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_83 = CEVAPOINT OF CENTROID AND SYMMEDIAN POINT (0.5865, 0.2047) X_111 = PARRY POINT (0.9958, -0.0649) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6571, 0.2840) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_755 = EVEN (0, 4) CIRCUMCIRCLE POINT (0.9607, -0.1944) X_32 = 3rd POWER POINT (0.6819, 0.3119) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (0.9964, 0.0602) X_31 = 2nd POWER POINT (0.6667, 0.2778) X_745 = EVEN (- 1, 3) CIRCUMCIRCLE POINT (0.9968, -0.0568) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6285, 0.2456) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (0.9529, -0.2118) X_827 = o-(0, 4) CIRCUMCIRCLE POINT (0.4499, 0.4975)
A
B
C
X826
X5
X83
X111
X251
X6
X755
X32
X733
X31
X745
X82
X759
X827
(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) { 5, 83 }: Central Line L(5, 83) { 111, 251 }: Central Line L(111, 251) { 6, 755 }: Central Line L(6, 755) { 32, 733 }: Central Line L(32, 733) { 31, 745 }: Central Line L(31, 745) { 82, 759 }: Central Line L(82, 759)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_826 = ODD (0, 4) INFINITY POINT (1.1069, 0.0700) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_83 = CEVAPOINT OF CENTROID AND SYMMEDIAN POINT (0.5729, 0.2075) X_111 = PARRY POINT (0.9823, -0.1319) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6328, 0.2886) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_755 = EVEN (0, 4) CIRCUMCIRCLE POINT (0.9295, -0.2560) X_32 = 3rd POWER POINT (0.6545, 0.3180) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0000, 0.0000) X_31 = 2nd POWER POINT (0.6411, 0.2822) X_745 = EVEN (- 1, 3) CIRCUMCIRCLE POINT (0.9843, -0.1244) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6084, 0.2491) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (0.9206, -0.2704) X_827 = o-(0, 4) CIRCUMCIRCLE POINT (0.4586, 0.4983)
A
B
C
X826
X5
X83
X111
X251
X6
X755
X32
X733
X31
X745
X82
X759
X827
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) { 5, 83 }: Central Line L(5, 83) { 111, 251 }: Central Line L(111, 251) { 6, 755 }: Central Line L(6, 755) { 32, 733 }: Central Line L(32, 733) { 31, 745 }: Central Line L(31, 745) { 82, 759 }: Central Line L(82, 759)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_826 = ODD (0, 4) INFINITY POINT (1.1992, 0.0530) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_83 = CEVAPOINT OF CENTROID AND SYMMEDIAN POINT (0.5905, 0.1374) X_111 = PARRY POINT (1.0593, -0.1621) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6603, 0.2039) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_755 = EVEN (0, 4) CIRCUMCIRCLE POINT (1.0603, -0.3818) X_32 = 3rd POWER POINT (0.6767, 0.2196) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (0.9896, 0.0181) X_31 = 2nd POWER POINT (0.6747, 0.1958) X_745 = EVEN (- 1, 3) CIRCUMCIRCLE POINT (1.0527, -0.1332) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6382, 0.1738) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (1.0381, -0.4642) X_827 = o-(0, 4) CIRCUMCIRCLE POINT (0.5640, 0.2922)
A
B
C
X826
X5
X83
X111
X251
X6
X755
X32
X733
X31
X745
X82
X759
X827
Kimberling (6, 9, 13) | AB = 1
X827 (o-(0, 4) CIRCUMCIRCLE POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[a^{-1}(b^{4} - c^{4}) + a^{3}(b^{0} - c^{0})] : :\)
Search Value (6, 9, 13): \(0.419571248227\)
X827
▸ Incident Central Lines12 lines
{5 83 6 755 31 745 32 733 82 759 111 251 163 813 249 805 250 935 560 719 662 831 741 849
▸ Functional Relations14 relations
Notes
Let A' , B' , C' be the intersections of line X23 X385 BC , CA , AB , resp. The circumcircles of AB'C' , BC'A' , CA'B' concur in X827 X23 X385 X2 Randy Hutson , December 10, 2016)
