X3111
ORTHOGONAL PROJECTION OF X2 ON BROCARD AXIS
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) { 3, 6 }: Brocard Axis { 2, 512 }: Central Line L(2, 512) { 373, 1316 }: Central Line L(373, 1316)
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) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2 = CENTROID (0.5374, 0.2000) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5614, 0.2395) X_1316 = 5th MOSES INTERSECTION (0.5678, 0.2795) X_3111 = ORTHOGONAL PROJECTION OF X(2) ON BROCARD AXIS (0.5534, 0.1903)
A
B
C
X3
X6
X2
X512
X373
X1316
X3111
(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, 6 }: Brocard Axis { 2, 512 }: Central Line L(2, 512) { 373, 1316 }: Central Line L(373, 1316)
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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_2 = CENTROID (0.5467, 0.1600) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5778, 0.2133) X_1316 = 5th MOSES INTERSECTION (0.5755, 0.2588) X_3111 = ORTHOGONAL PROJECTION OF X(2) ON BROCARD AXIS (0.5815, 0.1397)
A
B
C
X3
X6
X2
X512
X373
X1316
X3111
(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, 6 }: Brocard Axis { 2, 512 }: Central Line L(2, 512) { 373, 1316 }: Central Line L(373, 1316)
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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_2 = CENTROID (0.5393, 0.1620) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5656, 0.2159) X_1316 = 5th MOSES INTERSECTION (0.5623, 0.2565) X_3111 = ORTHOGONAL PROJECTION OF X(2) ON BROCARD AXIS (0.5712, 0.1465)
A
B
C
X3
X6
X2
X512
X373
X1316
X3111
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) { 3, 6 }: Brocard Axis { 2, 512 }: Central Line L(2, 512) { 373, 1316 }: Central Line L(373, 1316)
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) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_2 = CENTROID (0.5444, 0.0934) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5671, 0.1569) X_1316 = 5th MOSES INTERSECTION (0.5546, 0.1779) X_3111 = ORTHOGONAL PROJECTION OF X(2) ON BROCARD AXIS (0.6217, 0.0657)
A
B
C
X3
X6
X2
X512
X373
X1316
X3111
Kimberling (6, 9, 13) | AB = 1
X3111 (ORTHOGONAL PROJECTION OF X2 ON BROCARD AXIS ) 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 \(f(a,b,c) = a^{3}(b^{2} + c^{2} - a^{2})(b^{4} + c^{4} - 4b^{2}c^{2}) + ab^{2}c^{2}(b^{4} + c^{4})\)
Search Value (6, 9, 13): \(2.305148361376\)
▸ Incident Central Lines3 lines
{2 512 3 6 373 1316
▸ Functional Relations2 relations
X3111 X99 X2
X3111 X3
Notes
Let Ma Mb Mc be the medial triangle . Let A″ be the reflection of X115 Mb Mc , and define B″ and C″ cyclically. X3111 A″B″C″ .
