X2553
EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X4 ,2R))
The circle (
X4 ,2R) is identified at
X2474 .
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) { 2, 3 }: Euler Line { 388, 2463 }: Central Line L(388, 2463) { 497, 2464 }: Central Line L(497, 2464) { 1587, 2465 }: Central Line L(1587, 2465) { 1588, 2466 }: Central Line L(1588, 2466) { 2469, 2542 }: Central Line L(2469, 2542)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_388 = ∩X(7)X(8) (0.5790, 0.2635) X_2463 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5738, 0.2778) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.5593, 0.1851) X_2464 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5776, 0.3505) X_1587 = POINT CASTOR I (0.6026, 0.2943) X_2465 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.5818, 0.2893) X_1588 = POINT CASTOR II (0.5955, 0.2203) X_2466 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.5591, 0.3170) X_2469 = INSIMILICENTER(ORTHOCENTROIDALCIRCLE, BROCARD CIRCLE) (0.5619, 0.2389) X_2542 = INSIMILICENTER(BROCARD CIRCLE, (X(4),2R)) (0.5554, 0.2029) X_2553 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X(4),2R)) (0.5705, 0.2867)
A
B
C
X2
X3
X4
X388
X2463
X497
X2464
X1587
X2465
X1588
X2466
X2469
X2542
X2553
(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) { 2, 3 }: Euler Line { 388, 2463 }: Central Line L(388, 2463) { 497, 2464 }: Central Line L(497, 2464) { 1587, 2465 }: Central Line L(1587, 2465) { 1588, 2466 }: Central Line L(1588, 2466) { 2469, 2542 }: Central Line L(2469, 2542)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_4 = ORTHOCENTER (0.6400, 0.4800) X_388 = ∩X(7)X(8) (0.6067, 0.2467) X_2463 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5964, 0.2655) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.5900, 0.1300) X_2464 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5600, 0.9200) X_1587 = POINT CASTOR I (0.6400, 0.2865) X_2465 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.6125, 0.2872) X_1588 = POINT CASTOR II (0.6400, 0.1642) X_2466 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.4873, 0.5018) X_2469 = INSIMILICENTER(ORTHOCENTROIDALCIRCLE, BROCARD CIRCLE) (0.5806, 0.2109) X_2542 = INSIMILICENTER(BROCARD CIRCLE, (X(4),2R)) (0.5785, 0.1639) X_2553 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X(4),2R)) (0.5840, 0.2880)
A
B
C
X2
X3
X4
X388
X2463
X497
X2464
X1587
X2465
X1588
X2466
X2469
X2542
X2553
(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) { 2, 3 }: Euler Line { 388, 2463 }: Central Line L(388, 2463) { 497, 2464 }: Central Line L(497, 2464) { 1587, 2465 }: Central Line L(1587, 2465) { 1588, 2466 }: Central Line L(1588, 2466) { 2469, 2542 }: Central Line L(2469, 2542)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_4 = ORTHOCENTER (0.6180, 0.4859) X_388 = ∩X(7)X(8) (0.5898, 0.2498) X_2463 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5811, 0.2689) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.5755, 0.1302) X_2464 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5534, 0.8970) X_1587 = POINT CASTOR I (0.6180, 0.2904) X_2465 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.5947, 0.2910) X_1588 = POINT CASTOR II (0.6180, 0.1650) X_2466 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.4927, 0.5009) X_2469 = INSIMILICENTER(ORTHOCENTROIDALCIRCLE, BROCARD CIRCLE) (0.5678, 0.2121) X_2542 = INSIMILICENTER(BROCARD CIRCLE, (X(4),2R)) (0.5660, 0.1648) X_2553 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X(4),2R)) (0.5708, 0.2915)
A
B
C
X2
X3
X4
X388
X2463
X497
X2464
X1587
X2465
X1588
X2466
X2469
X2542
X2553
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) { 2, 3 }: Euler Line { 388, 2463 }: Central Line L(388, 2463) { 497, 2464 }: Central Line L(497, 2464) { 1587, 2465 }: Central Line L(1587, 2465) { 1588, 2466 }: Central Line L(1588, 2466) { 2469, 2542 }: Central Line L(2469, 2542)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2 = CENTROID (0.5444, 0.0934) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_4 = ORTHOCENTER (0.6331, 0.8294) X_388 = ∩X(7)X(8) (0.6172, 0.2016) X_2463 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.6085, 0.2161) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.6131, 0.0401) X_2464 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.6298, -0.0490) X_1587 = POINT CASTOR I (0.6539, 0.2598) X_2465 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.6342, 0.2653) X_1588 = POINT CASTOR II (0.6621, 0.0338) X_2466 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 2nd LEMOINE CIRCLE) (0.7285, -0.1417) X_2469 = INSIMILICENTER(ORTHOCENTROIDALCIRCLE, BROCARD CIRCLE) (0.5826, 0.1449) X_2542 = INSIMILICENTER(BROCARD CIRCLE, (X(4),2R)) (0.5879, 0.0957) X_2553 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X(4),2R)) (0.5674, 0.2843)
A
B
C
X2
X3
X4
X388
X2463
X497
X2464
X1587
X2465
X1588
X2466
X2469
X2542
X2553
Kimberling (6, 9, 13) | AB = 1
X2553 (EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X4 ,2R)) ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\) , where
\(f(A,B,C) = \cos A + (4 - J) \cos B \cos C\) , J as at
X1113 Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(0.475007060381\)
▸ Incident Central Lines11 lines
{2 3 388 2463 497 2464 1587 2465 1588 2466 2467 2550 2468 2551 2469 2542 2470 2543 2471 2546 2472 2547
Notes
As a point on the Euler line, X2553 \((1, -J + 3)\) .
