X3407

CSC(A + 2ω) POINT

X3407 and related points occupy Bernard Gibert’s Table 38. See the note at X3364.
< 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) { 76, 384 }: Central Line L(76, 384) { 1, 1501 }: Central Line L(1, 1501) { 6, 1916 }: Central Line L(6, 1916) { 983, 985 }: Central Line L(983, 985) { 321, 2205 }: Central Line L(321, 2205) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3094 = INTERSECTION OF LINES X(2)X(694) AND X(3)X(6) (0.5556, 0.1938) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_384 = THE CONWAY POINT: EULER LINE INTERCEPT OF LINE X(32)X(76) (0.5532, 0.2412) X_1 = INCENTER (0.5714, 0.2333) X_1501 = TRILINEAR 5th POWER POINT (0.6596, 0.3922) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_1916 = TRILINEAR POLE OF PU(11) (1.0447, 0.0288) X_983 = ISOGONAL CONJUGATE OF X(982) (0.5890, 0.2731) X_985 = ISOGONAL CONJUGATE OF X(984) (0.5961, 0.2693) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4800, 0.1527) X_2205 = -ISOCONJUGATE OF X(310) (0.6477, 0.3479) X_3407 = CSC(A + 2ω) POINT (0.5852, 0.2751) A B C X3094 X76 X384 X1 X1501 X6 X1916 X983 X985 X321 X2205 X3407 (a, b, c) = (5, 6, 7) | AB = 1

X3407 (CSC(A + 2ω) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\csc(A + 2\omega )\)
Search Value (6, 9, 13): \(1.145037365826\)

X3407 lies on the Kiepert hyperbola.

▸ Incident Central Lines5 lines

{1, 1501}, {6, 1916}, {76, 384}, {321, 2205}, {983, 985}

▸ Functional Relations3 relations

Notes

Let A'B'C' be the 1st Brocard triangle. X3407 is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, July 20, 2016)

Let A'B'C' be the 1st Brocard triangle and A″B″C″ be the 1st anti-Brocard triangle. Let A* be the diagonal crosspoint of trapezoid B’C'C″B″ (i.e., the intersection of lines B'C″ and C'B″); define B* and C* cyclically. The lines AA*, BB*, CC* concur in X3407. (Randy Hutson, July 20, 2016)