X251
ISOGONAL CONJUGATE OF X141
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 1369 }: Central Line L(2, 1369) { 6, 22 }: Central Line L(6, 22) { 2, 32 }: Central Line L(2, 32) { 37, 82 }: Central Line L(37, 82) { 184, 263 }: Central Line L(184, 263)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_1369 = ANTICOMPLEMENTARY CONJUGATE OF X(6) (0.3769, -0.0086) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_22 = EXETER POINT (0.4143, -0.1221) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_82 = ISOGONAL CONJUGATE OF X(38) (0.5943, 0.2703) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_263 = ISOGONAL CONJUGATE OF X(183) (0.6298, 0.3525) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6177, 0.3042)
A
B
C
Ma
Mb
Mc
X2
X1369
X6
X22
X32
X37
X82
X184
X263
X251
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 1369 }: Central Line L(2, 1369) { 6, 22 }: Central Line L(6, 22) { 2, 32 }: Central Line L(2, 32) { 37, 82 }: Central Line L(37, 82) { 184, 263 }: Central Line L(184, 263)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) M_a = Midpoint M_a of BC (0.8200, 0.2400) M_b = Midpoint M_b of CA (0.3200, 0.2400) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5467, 0.1600) X_1369 = ANTICOMPLEMENTARY CONJUGATE OF X(6) (0.3258, -0.0880) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_22 = EXETER POINT (0.3600, -0.4800) X_32 = 3rd POWER POINT (0.6819, 0.3119) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5787, 0.1787) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6285, 0.2456) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.6400, 0.0000) X_263 = ISOGONAL CONJUGATE OF X(183) (0.6615, 0.3569) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6571, 0.2840)
A
B
C
Ma
Mb
Mc
X2
X1369
X6
X22
X32
X37
X82
X184
X263
X251
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 1369 }: Central Line L(2, 1369) { 6, 22 }: Central Line L(6, 22) { 2, 32 }: Central Line L(2, 32) { 37, 82 }: Central Line L(37, 82) { 184, 263 }: Central Line L(184, 263)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) M_a = Midpoint M_a of BC (0.8090, 0.2429) M_b = Midpoint M_b of CA (0.3090, 0.2429) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5393, 0.1620) X_1369 = ANTICOMPLEMENTARY CONJUGATE OF X(6) (0.3524, -0.0914) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_22 = EXETER POINT (0.3820, -0.4859) X_32 = 3rd POWER POINT (0.6545, 0.3180) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5661, 0.1805) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6084, 0.2491) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.6180, 0.0000) X_263 = ISOGONAL CONJUGATE OF X(183) (0.6366, 0.3621) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6328, 0.2886)
A
B
C
Ma
Mb
Mc
X2
X1369
X6
X22
X32
X37
X82
X184
X263
X251
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 1369 }: Central Line L(2, 1369) { 6, 22 }: Central Line L(6, 22) { 2, 32 }: Central Line L(2, 32) { 37, 82 }: Central Line L(37, 82) { 184, 263 }: Central Line L(184, 263)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) M_a = Midpoint M_a of BC (0.8166, 0.1400) M_b = Midpoint M_b of CA (0.3166, 0.1400) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5444, 0.0934) X_1369 = ANTICOMPLEMENTARY CONJUGATE OF X(6) (0.3126, -0.1278) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_22 = EXETER POINT (0.8328, 2.4854) X_32 = 3rd POWER POINT (0.6767, 0.2196) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5913, 0.1097) X_82 = ISOGONAL CONJUGATE OF X(38) (0.6382, 0.1738) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.3215, 1.0549) X_263 = ISOGONAL CONJUGATE OF X(183) (0.6200, 0.3050) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6603, 0.2039)
A
B
C
Ma
Mb
Mc
X2
X1369
X6
X22
X32
X37
X82
X184
X263
X251
Kimberling (6, 9, 13) | AB = 1
X251 (ISOGONAL CONJUGATE OF X141 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a^{2}\csc(A + \omega ) : b^{2}\csc(B + \omega ) : c^{2}\csc(C + \omega )\)
Barycentrics: \(a^{3}\csc(A + \omega ) : b^{3}\csc(B + \omega ) : c^{3}\csc(C + \omega )\)
Search Value (6, 9, 13): \(0.572649002745\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines9 lines
{2 32 6 22 37 82 110 694 112 427 184 263 308 385 609 614 689 699
▸ Functional Relations13 relations
Notes
Let K be the symmedian point of ABC and let A' be the symmedian point of the triangle BCK ; define B' and C' cyclically. The lines AA' , BB' , CC' concur in X251 Randy Hutson , 9/23/2011)
Let A5 'B5 'C5 ' be the 5th anti-Brocard triangle. The radical center of the circumcircles of BCA5 ' , CAB5 ' , ABC5 ' is X251
