X2556
1st INTERSECTION(LINE X1 X_3, 2nd BROCARD CIRCLE)
Of the two points of intersection,
X2556 is one closer to
X1 .
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) { 2557, 3 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2557 = 2nd INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.4892, 0.0822) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_2556 = 1st INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.5108, 0.1220)
A
B
C
X2557
X3
X1
X2556
(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) { 2557, 3 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2557 = 2nd INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.4800, -0.0401) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1 = INCENTER (0.6000, 0.2000) X_2556 = 1st INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.5200, 0.0401)
A
B
C
X2557
X3
X1
X2556
(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) { 2557, 3 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2557 = 2nd INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.4495, -0.1214) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1 = INCENTER (0.5841, 0.2021) X_2556 = 1st INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.5505, 0.1214)
A
B
C
X2557
X3
X1
X2556
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) { 2557, 3 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2557 = 2nd INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.4938, -0.2965) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_1 = INCENTER (0.6154, 0.1300) X_2556 = 1st INTERSECTION(LINE X(1)X(3), 2nd BROCARD CIRCLE) (0.5062, -0.2529)
A
B
C
X2557
X3
X1
X2556
Kimberling (6, 9, 13) | AB = 1
X2556 (1st INTERSECTION(LINE X1 X_3, 2nd BROCARD CIRCLE) ) 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
\(f(A,B,C) = 1 - \cos B - \cos C - (d/e) \cos A\) , where
\(d = (3 - 2 \cos A - 2 \cos B - 2 \cos C)^{1/2} = |IO|/R = 1 - 2r/R\) , where I =
X1 ,
O =
X3 .,
\(e = (1 - 4 \sin^{2}\omega )^{1/2}\) Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(6.508377450098\)
▸ Incident Central Lines1 line
▸ Functional Relations1 relation
