X593
1st HATZIPOLAKIS-YIU POINT
Let O(A) be the circle tangent to line
BC and to the circumcircle of triangle
ABC at vertex
A . Let
AB and
AC be where O(A) meets lines
AB and
AC , respectively. Let L(A) be the line joining
AB and
AC , and define L(B) and L(C) cyclically. Let
A' be where L(B) and L(C) meet, and define
B' and
C' cyclically. Then triangle
A'B'C' is homothetic to triangle
ABC , and the center of homothety is
X593 . See
Antreas Hatzipolakis ,
Paul Yiu ,
Hyacinthos #2070 .
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) { 36, 58 }: Central Line L(36, 58) { 31, 110 }: Central Line L(31, 110) { 2, 261 }: Central Line L(2, 261) { 81, 757 }: Central Line L(81, 757) { 115, 1029 }: Central Line L(115, 1029) { 229, 1104 }: Central Line L(229, 1104)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_594 = ISOGONAL CONJUGATE OF X(593) (0.5025, 0.1672) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_2 = CENTROID (0.5374, 0.2000) X_261 = ISOTOMIC CONJUGATE OF X(12) (0.5072, 0.1625) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_757 = ISOGONAL CONJUGATE OF X(756) (0.5971, 0.2688) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_1029 = CYCLOCEVIAN CONJUGATE OF X(1) (0.6026, 0.3320) X_229 = -CEVA CONJUGATE OF X(81) (0.5961, 0.2991) X_1104 = CROSSPOINT OF X(1) AND X(28) (0.6311, 0.3042) X_593 = 1st HATZIPOLAKIS-YIU POINT (0.6202, 0.3026)
A
B
C
X594
X36
X58
X31
X110
X2
X261
X81
X757
X115
X1029
X229
X1104
X593
(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) { 36, 58 }: Central Line L(36, 58) { 31, 110 }: Central Line L(31, 110) { 2, 261 }: Central Line L(2, 261) { 81, 757 }: Central Line L(81, 757) { 115, 1029 }: Central Line L(115, 1029) { 229, 1104 }: Central Line L(229, 1104)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4915, 0.1212) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.0000, 1.0000) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6522, 0.2609) X_31 = 2nd POWER POINT (0.6667, 0.2778) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_261 = ISOTOMIC CONJUGATE OF X(12) (0.4996, 0.1104) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.6185, 0.2215) X_757 = ISOGONAL CONJUGATE OF X(756) (0.6340, 0.2430) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7445, 0.0609) X_1029 = CYCLOCEVIAN CONJUGATE OF X(1) (0.6339, 0.3549) X_229 = -CEVA CONJUGATE OF X(81) (0.6243, 0.2912) X_1104 = CROSSPOINT OF X(1) AND X(28) (0.6769, 0.2769) X_593 = 1st HATZIPOLAKIS-YIU POINT (0.6617, 0.2811)
A
B
C
X594
X36
X58
X31
X110
X2
X261
X81
X757
X115
X1029
X229
X1104
X593
(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) { 36, 58 }: Central Line L(36, 58) { 31, 110 }: Central Line L(31, 110) { 2, 261 }: Central Line L(2, 261) { 81, 757 }: Central Line L(81, 757) { 115, 1029 }: Central Line L(115, 1029) { 229, 1104 }: Central Line L(229, 1104)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4931, 0.1231) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.9387, 1.0546) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6285, 0.2645) X_31 = 2nd POWER POINT (0.6411, 0.2822) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_261 = ISOTOMIC CONJUGATE OF X(12) (0.5000, 0.1120) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5997, 0.2241) X_757 = ISOGONAL CONJUGATE OF X(756) (0.6130, 0.2461) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7135, 0.0464) X_1029 = CYCLOCEVIAN CONJUGATE OF X(1) (0.6129, 0.3600) X_229 = -CEVA CONJUGATE OF X(81) (0.6047, 0.2952) X_1104 = CROSSPOINT OF X(1) AND X(28) (0.6501, 0.2815) X_593 = 1st HATZIPOLAKIS-YIU POINT (0.6368, 0.2855)
A
B
C
X594
X36
X58
X31
X110
X2
X261
X81
X757
X115
X1029
X229
X1104
X593
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) { 36, 58 }: Central Line L(36, 58) { 31, 110 }: Central Line L(31, 110) { 2, 261 }: Central Line L(2, 261) { 81, 757 }: Central Line L(81, 757) { 115, 1029 }: Central Line L(115, 1029) { 229, 1104 }: Central Line L(229, 1104)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4705, 0.0589) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.7120, 0.4690) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6639, 0.1838) X_31 = 2nd POWER POINT (0.6747, 0.1958) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_2 = CENTROID (0.5444, 0.0934) X_261 = ISOTOMIC CONJUGATE OF X(12) (0.4785, 0.0357) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.6338, 0.1505) X_757 = ISOGONAL CONJUGATE OF X(756) (0.6468, 0.1701) X_115 = CENTER OF KIEPERT HYPERBOLA (0.6909, 0.0207) X_1029 = CYCLOCEVIAN CONJUGATE OF X(1) (0.6403, 0.3963) X_229 = -CEVA CONJUGATE OF X(81) (0.6164, 0.2352) X_1104 = CROSSPOINT OF X(1) AND X(28) (0.6842, 0.1882) X_593 = 1st HATZIPOLAKIS-YIU POINT (0.6667, 0.2004)
A
B
C
X594
X36
X58
X31
X110
X2
X261
X81
X757
X115
X1029
X229
X1104
X593
Kimberling (6, 9, 13) | AB = 1
X593 (1st HATZIPOLAKIS-YIU POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a/(b + c)^{2} : b/(c + a)^{2} : c/(a + b)^{2}\)
Barycentrics: \([a/(b + c)]^{2} : [b/(c + a)]^{2} : [c/(a + b)]^{2}\)
Search Value (6, 9, 13): \(0.558849923867\)
▸ Incident Central Lines6 lines
{2 261 31 110 36 58 81 757 115 1029 229 1104
▸ Functional Relations4 relations
