X1658
CIRCUMCENTER OF KOSNITA TRIANGLE
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC < O_A O_B O_C >: Kosnita Triangle △O_A O_B O_C
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle_of_kosnita: Circumcircle of kosnita [ O_A X_1658 ]: Line [ O_A X_1658 ] [ O_B X_1658 ]: Line [ O_B X_1658 ] [ O_C X_1658 ]: Line [ O_C X_1658 ] { 2, 3 }: Euler Line { 54, 568 }: Central Line L(54, 568) { 143, 578 }: Central Line L(143, 578) { 569, 973 }: Central Line L(569, 973) { 1147, 1154 }: Central Line L(1147, 1154)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) O_A = Kosnita Circumcenter O_A (0.8000, 0.2960) O_B = Kosnita Circumcenter O_B (0.1711, 0.4378) O_C = Kosnita Circumcenter O_C (0.5000, -1.1737) X_2 = CENTROID (0.5374, 0.2000) X_54 = KOSNITA POINT (0.5880, 0.2176) X_568 = ORTHOHARMONIC OF X(50) (0.6437, 0.3393) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_578 = ORTHOHARMONIC OF X(216) (0.5933, 0.2561) X_569 = HARMONIC OF X(52) (0.5651, 0.2096) X_973 = 1st EHRMANN POINT (0.6138, 0.3246) X_1147 = ISOGONAL CONJUGATE OF X(847) (0.8329, 0.3027) X_1154 = ISOGONAL CONJUGATE OF X(1141) (0.8932, 0.6667) X_1658 = CIRCUMCENTER OF KOSNITA TRIANGLE (0.3191, -0.3713)
A
B
C
X3
OA
OB
OC
X2
X54
X568
X143
X578
X569
X973
X1147
X1154
X1658
(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) { 3, 26 }: Euler Line { 2, 3 }: Euler Line { 54, 568 }: Central Line L(54, 568) { 143, 578 }: Central Line L(143, 578) { 569, 973 }: Central Line L(569, 973) { 1147, 1154 }: Central Line L(1147, 1154)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_26 = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE (0.7077, 0.7120) X_2 = CENTROID (0.5467, 0.1600) X_54 = KOSNITA POINT (0.6400, 0.1600) X_568 = ORTHOHARMONIC OF X(50) (0.6867, 0.3200) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.7120, 0.2331) X_578 = ORTHOHARMONIC OF X(216) (0.6400, 0.2400) X_569 = HARMONIC OF X(52) (0.5933, 0.1600) X_973 = 1st EHRMANN POINT (0.6400, 0.3200) X_1147 = ISOGONAL CONJUGATE OF X(847) (-0.0283, 0.1600) X_1154 = ISOGONAL CONJUGATE OF X(1141) (0.9253, 0.5927) X_1658 = CIRCUMCENTER OF KOSNITA TRIANGLE (0.3857, -0.3920)
A
B
C
X3
X26
X2
X54
X568
X143
X578
X569
X973
X1147
X1154
X1658
(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) { 3, 26 }: Euler Line { 2, 3 }: Euler Line { 54, 568 }: Central Line L(54, 568) { 143, 578 }: Central Line L(143, 578) { 569, 973 }: Central Line L(569, 973) { 1147, 1154 }: Central Line L(1147, 1154)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_26 = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE (0.6751, 0.7207) X_2 = CENTROID (0.5393, 0.1620) X_54 = KOSNITA POINT (0.6180, 0.1620) X_568 = ORTHOHARMONIC OF X(50) (0.6574, 0.3239) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6491, 0.3580) X_578 = ORTHOHARMONIC OF X(216) (0.6180, 0.2429) X_569 = HARMONIC OF X(52) (0.5787, 0.1620) X_973 = 1st EHRMANN POINT (0.6180, 0.3239) X_1147 = ISOGONAL CONJUGATE OF X(847) (-0.0357, 0.1620) X_1154 = ISOGONAL CONJUGATE OF X(1141) (0.8780, 0.6266) X_1658 = CIRCUMCENTER OF KOSNITA TRIANGLE (0.4036, -0.3968)
A
B
C
X3
X26
X2
X54
X568
X143
X578
X569
X973
X1147
X1154
X1658
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC < O_A O_B O_C >: Kosnita Triangle △O_A O_B O_C
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle_of_kosnita: Circumcircle of kosnita [ O_A X_1658 ]: Line [ O_A X_1658 ] [ O_B X_1658 ]: Line [ O_B X_1658 ] [ O_C X_1658 ]: Line [ O_C X_1658 ] { 2, 3 }: Euler Line { 54, 568 }: Central Line L(54, 568) { 143, 578 }: Central Line L(143, 578) { 569, 973 }: Central Line L(569, 973) { 1147, 1154 }: Central Line L(1147, 1154)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3 = CIRCUMCENTER (0.5000, -0.2747) O_A = Kosnita Circumcenter O_A (0.6893, -0.0268) O_B = Kosnita Circumcenter O_B (0.3548, 0.0535) O_C = Kosnita Circumcenter O_C (0.5000, 0.3177) X_2 = CENTROID (0.5444, 0.0934) X_54 = KOSNITA POINT (1.8800, -0.0730) X_568 = ORTHOHARMONIC OF X(50) (0.6364, 0.1068) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6356, 0.2874) X_578 = ORTHOHARMONIC OF X(216) (0.8409, 0.6790) X_569 = HARMONIC OF X(52) (0.6278, 0.0830) X_973 = 1st EHRMANN POINT (0.2905, 0.2370) X_1147 = ISOGONAL CONJUGATE OF X(847) (0.4490, 0.1052) X_1154 = ISOGONAL CONJUGATE OF X(1141) (1.1935, 0.1882) X_1658 = CIRCUMCENTER OF KOSNITA TRIANGLE (0.5475, 0.1196)
A
B
C
X3
OA
OB
OC
X2
X54
X568
X143
X578
X569
X973
X1147
X1154
X1658
Kimberling (6, 9, 13) | AB = 1
X1658 (CIRCUMCENTER OF KOSNITA TRIANGLE ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Constructed Elements
Derived Triangle: Kosnita Triangle △OA OB OC — vertices OA , OB , OC (Circumcenters of △BOC, △COA, △AOB where O = X3 )
Circle: circumcircleof kosnita — Centered at X1658
Concurrence Lines: OA –X1658 , OB –X1658 , OC –X1658 concurring at X1658
Central Lines: {2, 3} (Euler line )
Related Centers: X3 (CIRCUMCENTER X2 (CENTROID
Trilinears: \(\cos(B - C) + 4 \cos 2A \cos B \cos C : :\) (Nikolaos Dergiades,
Hyacinthos 7752)
Search Value (6, 9, 13): \(2.332793429859\)
▸ Incident Central Lines6 lines
{2 3 54 568 143 578 569 973 1092 1511 1147 1154
▸ Functional Relations1 relation
Notes
As a point on the Euler line, X1658 \((E + 8F, -3E - 8F)\) .
Let La be the polar of X4 A and passing through X3 Lb , Lc cyclically. (Note: X4 ABC .) Let A″ = Lb ∩Lc , and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC , and its nine-point center is X1658 Randy Hutson , July 20, 2016)
