X825
o-(0, 3) CIRCUMCIRCLE POINT
X825 = intersection, other than
A ,
B ,
C , of circumcircle and hyperbola {
A ,
B ,
C ,PU(12)}}
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) { 99, 163 }: Central Line L(99, 163) { 103, 572 }: Central Line L(103, 572) { 6, 753 }: Central Line L(6, 753) { 1, 761 }: Central Line L(1, 761) { 32, 731 }: Central Line L(32, 731) { 31, 743 }: Central Line L(31, 743)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_824 = ODD (0, 3) INFINITY POINT (1.0936, 0.0129) X_99 = STEINER POINT (0.1109, 0.4322) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.3174, 0.5883) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_572 = ORTHOGONAL OF X(58) (7.2953, -0.6907) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_753 = EVEN (0, 3) CIRCUMCIRCLE POINT (0.9725, -0.0907) X_1 = INCENTER (0.5714, 0.2333) X_761 = EVEN (1, 4) CIRCUMCIRCLE POINT (0.9239, -0.1820) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_731 = EVEN (- 2, 1) CIRCUMCIRCLE POINT (1.0101, 0.0876) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_743 = EVEN (- 1, 2) CIRCUMCIRCLE POINT (1.0002, 0.0008) X_825 = o-(0, 3) CIRCUMCIRCLE POINT (0.2886, 0.5665)
A
B
C
X824
X99
X163
X103
X572
X6
X753
X1
X761
X32
X731
X31
X743
X825
(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) { 99, 163 }: Central Line L(99, 163) { 103, 572 }: Central Line L(103, 572) { 6, 753 }: Central Line L(6, 753) { 1, 761 }: Central Line L(1, 761) { 32, 731 }: Central Line L(32, 731) { 31, 743 }: Central Line L(31, 743)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_824 = ODD (0, 3) INFINITY POINT (1.1059, 0.0264) X_99 = STEINER POINT (0.1511, 0.3581) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.4407, 0.5085) X_103 = ANTIPODE OF X(101) (0.6923, -0.4615) X_572 = ORTHOGONAL OF X(58) (0.2356, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_753 = EVEN (0, 3) CIRCUMCIRCLE POINT (0.9825, -0.1310) X_1 = INCENTER (0.6000, 0.2000) X_761 = EVEN (1, 4) CIRCUMCIRCLE POINT (0.9154, -0.2783) X_32 = 3rd POWER POINT (0.6819, 0.3119) X_731 = EVEN (- 2, 1) CIRCUMCIRCLE POINT (0.9874, 0.1117) X_31 = 2nd POWER POINT (0.6667, 0.2778) X_743 = EVEN (- 1, 2) CIRCUMCIRCLE POINT (1.0000, 0.0026) X_825 = o-(0, 3) CIRCUMCIRCLE POINT (0.4079, 0.4914)
A
B
C
X824
X99
X163
X103
X572
X6
X753
X1
X761
X32
X731
X31
X743
X825
(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) { 99, 163 }: Central Line L(99, 163) { 103, 572 }: Central Line L(103, 572) { 6, 753 }: Central Line L(6, 753) { 1, 761 }: Central Line L(1, 761) { 32, 731 }: Central Line L(32, 731) { 31, 743 }: Central Line L(31, 743)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_824 = ODD (0, 3) INFINITY POINT (1.1033, 0.0497) X_99 = STEINER POINT (0.1910, 0.3931) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.4503, 0.5059) X_103 = ANTIPODE OF X(101) (0.6635, -0.4725) X_572 = ORTHOGONAL OF X(58) (0.6858, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_753 = EVEN (0, 3) CIRCUMCIRCLE POINT (0.9597, -0.1967) X_1 = INCENTER (0.5841, 0.2021) X_761 = EVEN (1, 4) CIRCUMCIRCLE POINT (0.8769, -0.3285) X_32 = 3rd POWER POINT (0.6545, 0.3180) X_731 = EVEN (- 2, 1) CIRCUMCIRCLE POINT (0.9966, 0.0580) X_31 = 2nd POWER POINT (0.6411, 0.2822) X_743 = EVEN (- 1, 2) CIRCUMCIRCLE POINT (0.9961, -0.0626) X_825 = o-(0, 3) CIRCUMCIRCLE POINT (0.4229, 0.4940)
A
B
C
X824
X99
X163
X103
X572
X6
X753
X1
X761
X32
X731
X31
X743
X825
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) { 99, 163 }: Central Line L(99, 163) { 103, 572 }: Central Line L(103, 572) { 6, 753 }: Central Line L(6, 753) { 1, 761 }: Central Line L(1, 761) { 32, 731 }: Central Line L(32, 731) { 31, 743 }: Central Line L(31, 743)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_824 = ODD (0, 3) INFINITY POINT (1.1980, 0.0367) X_99 = STEINER POINT (0.2513, 0.2387) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.5553, 0.2986) X_103 = ANTIPODE OF X(101) (0.5518, -0.8428) X_572 = ORTHOGONAL OF X(58) (1.1910, 0.5833) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_753 = EVEN (0, 3) CIRCUMCIRCLE POINT (1.0705, -0.2731) X_1 = INCENTER (0.6154, 0.1300) X_761 = EVEN (1, 4) CIRCUMCIRCLE POINT (0.9688, -0.5997) X_32 = 3rd POWER POINT (0.6767, 0.2196) X_731 = EVEN (- 2, 1) CIRCUMCIRCLE POINT (0.9523, 0.0730) X_31 = 2nd POWER POINT (0.6747, 0.1958) X_743 = EVEN (- 1, 2) CIRCUMCIRCLE POINT (1.0260, -0.0539) X_825 = o-(0, 3) CIRCUMCIRCLE POINT (0.5356, 0.2947)
A
B
C
X824
X99
X163
X103
X572
X6
X753
X1
X761
X32
X731
X31
X743
X825
Kimberling (6, 9, 13) | AB = 1
X825 (o-(0, 3) CIRCUMCIRCLE POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[a^{-1}(b^{3} - c^{3}) + a^{2}(b^{0} - c^{0})] : :\)
Search Value (6, 9, 13): \(0.617886287890\)
X825
▸ Incident Central Lines10 lines
{1 761 6 753 31 743 32 731 99 163 103 572 105 985 560 717 692 813 767 870
▸ Functional Relations8 relations
X825 isogonal conjugate of X824
X825 A , B , C , of circumcircle and hyperbola {A , B , C ,PU(12)}}
X825 trilinear pole of line X6 X560
X825 X1 X32
X825 X2 X31
X825 X6 X560
X825 X76 X1
X825 X2 X31
Notes
Let Q be a point on line X2 X31 X2 A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' ; equivalently, A″ = BC ∩B'C' / Define B″ and C″ cyclically. The circumcircles of AB″C″ , BC″A″ , CA″B″ concur in X825 Randy Hutson , December 29, 2015)
