X3644
REFLECTION OF X75 IN X192
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) { 75, 192 }: Central Line L(75, 192) { 2, 37 }: Central Line L(2, 37) { 190, 1743 }: Central Line L(190, 1743) { 726, 3244 }: Central Line L(726, 3244)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_192 = -CEVA CONJUGATE OF X(2) (CONGRUENT PARALLELIANS POINT) (0.6147, 0.2635) X_2 = CENTROID (0.5374, 0.2000) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_1743 = MIMOSA TRANSFORM OF X(222) (0.6798, 0.3620) X_726 = EVEN (- 2, -1) INFINITY POINT (1.0187, 0.5358) X_3244 = REFLECTION OF X(10) IN X(1) (0.6224, 0.2833) X_3644 = REFLECTION OF X(75) IN X(192) (0.7307, 0.3588)
A
B
C
X75
X192
X2
X37
X190
X1743
X726
X3244
X3644
(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) { 75, 192 }: Central Line L(75, 192) { 2, 37 }: Central Line L(2, 37) { 190, 1743 }: Central Line L(190, 1743) { 726, 3244 }: Central Line L(726, 3244)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4826, 0.1226) X_192 = -CEVA CONJUGATE OF X(2) (CONGRUENT PARALLELIANS POINT) (0.6749, 0.2349) X_2 = CENTROID (0.5467, 0.1600) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5787, 0.1787) X_190 = YFF PARABOLIC POINT (0.0933, 0.3200) X_1743 = MIMOSA TRANSFORM OF X(222) (0.7429, 0.3429) X_726 = EVEN (- 2, -1) INFINITY POINT (1.0679, 0.4028) X_3244 = REFLECTION OF X(10) IN X(1) (0.6800, 0.2600) X_3644 = REFLECTION OF X(75) IN X(192) (0.8672, 0.3472)
A
B
C
X75
X192
X2
X37
X190
X1743
X726
X3244
X3644
(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) { 75, 192 }: Central Line L(75, 192) { 2, 37 }: Central Line L(2, 37) { 190, 1743 }: Central Line L(190, 1743) { 726, 3244 }: Central Line L(726, 3244)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4859, 0.1249) X_192 = -CEVA CONJUGATE OF X(2) (CONGRUENT PARALLELIANS POINT) (0.6463, 0.2361) X_2 = CENTROID (0.5393, 0.1620) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5661, 0.1805) X_190 = YFF PARABOLIC POINT (0.1322, 0.3610) X_1743 = MIMOSA TRANSFORM OF X(222) (0.7065, 0.3493) X_726 = EVEN (- 2, -1) INFINITY POINT (1.0440, 0.4375) X_3244 = REFLECTION OF X(10) IN X(1) (0.6511, 0.2623) X_3644 = REFLECTION OF X(75) IN X(192) (0.8067, 0.3472)
A
B
C
X75
X192
X2
X37
X190
X1743
X726
X3244
X3644
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) { 75, 192 }: Central Line L(75, 192) { 2, 37 }: Central Line L(2, 37) { 190, 1743 }: Central Line L(190, 1743) { 726, 3244 }: Central Line L(726, 3244)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4506, 0.0607) X_192 = -CEVA CONJUGATE OF X(2) (CONGRUENT PARALLELIANS POINT) (0.7320, 0.1586) X_2 = CENTROID (0.5444, 0.0934) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5913, 0.1097) X_190 = YFF PARABOLIC POINT (0.1548, 0.2119) X_1743 = MIMOSA TRANSFORM OF X(222) (0.7487, 0.2427) X_726 = EVEN (- 2, -1) INFINITY POINT (1.1816, 0.2494) X_3244 = REFLECTION OF X(10) IN X(1) (0.7219, 0.1850) X_3644 = REFLECTION OF X(75) IN X(192) (1.0135, 0.2564)
A
B
C
X75
X192
X2
X37
X190
X1743
X726
X3244
X3644
Kimberling (6, 9, 13) | AB = 1
X3644 (REFLECTION OF X75 IN X192 ) 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 \(f(a,b,c) = bc(3/a - 2/b - 2/c)\)
Barycentrics: \(3/a - 2/b - 2/c : 3/b - 2/c - 2/a : 3/c - 2/a - 2/b\)
Search Value (6, 9, 13): \(-2.756085360801\)
▸ Linear Combinations1 combination
▸ Incident Central Lines3 lines
▸ Functional Relations2 relations
Notes
For any point P , let L(A) be the line through P parallel to BC . Let U = AB ∩L(A) and V = AC ∩L(A), and let a’ = |UV |. Define b’ and c’ cyclically. Let L be the line consisting of points P such that a + ta’ = b + tb’ = c + tc’. Then X3644 X75 X37 X192
