X98 –X112
Centers on the Circumcircle
Notable points on the circumcircle of ABC , including the Tarry point, Steiner point, anticomplementary, Feuerbach, Kiepert, and Jerabek antipodes, and centers derived from circumcircle mappings.
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_98 = TARRY POINT (0.8891, -0.2281) X_99 = STEINER POINT (0.1109, 0.4322) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_102 = Λ(INCENTER, ORTHOCENTER) (0.6364, -0.3897) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5733, 0.6071) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.5320, 0.6114) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_111 = PARRY POINT (0.9885, -0.0457) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121)
A
B
C
X3
X1
X98
X99
X100
X101
X102
X103
X104
X105
X106
X107
X108
X109
X110
X111
X112
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8000, 0.0000) R_a = Radius Point R_a (0.4000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1 = INCENTER (0.6000, 0.2000) X_98 = TARRY POINT (0.8489, -0.3581) X_99 = STEINER POINT (0.1511, 0.3581) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.3077, 0.4615) X_102 = Λ(INCENTER, ORTHOCENTER) (0.5000, -0.5000) X_103 = ANTIPODE OF X(101) (0.6923, -0.4615) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9800, -0.1400) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.6400, 0.4800) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.6400, 0.4800) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_111 = PARRY POINT (0.9958, -0.0649) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6400, 0.4800)
A
B
C
X3
X1
X98
X99
X100
X101
X102
X103
X104
X105
X106
X107
X108
X109
X110
X111
X112
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.7862, 0.0000) R_a = Radius Point R_a (0.3820, 0.0000) C = Vertex C (0.6180, 0.4859) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1 = INCENTER (0.5841, 0.2021) X_98 = TARRY POINT (0.8090, -0.3931) X_99 = STEINER POINT (0.1910, 0.3931) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.3365, 0.4725) X_102 = Λ(INCENTER, ORTHOCENTER) (0.5000, -0.5000) X_103 = ANTIPODE OF X(101) (0.6635, -0.4725) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9563, -0.2045) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.6180, 0.4859) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.6180, 0.4859) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_111 = PARRY POINT (0.9823, -0.1319) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6180, 0.4859)
A
B
C
X3
X1
X98
X99
X100
X101
X102
X103
X104
X105
X106
X107
X108
X109
X110
X111
X112
(a, b, c) = (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.6923, 0.0000) R_a = Radius Point R_a (0.5385, 0.0000) C = Vertex C (0.6331, 0.2801) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_1 = INCENTER (0.6154, 0.1300) X_98 = TARRY POINT (0.7487, -0.7881) X_99 = STEINER POINT (0.2513, 0.2387) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.4482, 0.2934) X_102 = Λ(INCENTER, ORTHOCENTER) (0.3952, -0.8354) X_103 = ANTIPODE OF X(101) (0.5518, -0.8428) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (1.0692, -0.3135) X_106 = Λ(INCENTER, CENTROID) (1.0294, -0.0621) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5910, 0.2885) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.6487, 0.2761) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_111 = PARRY POINT (1.0593, -0.1621) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6732, 0.2689)
A
B
C
X3
X1
X98
X99
X100
X101
X102
X103
X104
X105
X106
X107
X108
X109
X110
X111
X112
(a, b, c) = (6, 9, 13) | AB = 1
X98 –X112 (Centers on the Circumcircle ) — 15 centers constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Group Preamble — Centers on the circumcircle: X74 , X98 -X112 (Points on the circumcircle; mappings Λ and Ψ)
Centers X74 X98 X112 Λ and Ψ derived from such a point P for application to points X , are defined here:
Λ(P,X) = isogonal conjugate of the point where line PX meets the line at infinity.
Let Y = Λ(P,X), let Q = isogonal conjugate of P , and let Y and Z be the points where line YQ meets the circumcircle;
then Ψ(P,X) = Z.
X98 is the {X2 ,X147 }-harmonic conjugate of X114 . For a list of harmonic conjugates, click Tables at the top of this page.
X99 = X6 -of-1st-anti-Brocard-triangle
X100 is the anticomplement of Feuerbach point.
X101 = anticomplement of X116
X102 = anticomplement of X117
X103 is the antipode of X101 .
X104 is the antipode of X100 .
X105 = anticomplement of X120
X106 = anticomplement of X121
X107 = anticomplement of X122
X108 = anticomplement of X123
X109 = anticomplement of X124
Scimemi proves that if the Euler line is reflected in every side of triangle ABC , then the three reflections concur in X110 .
X111 = anticomplement of X126
X112 = anticomplement of X127
