X3406

SEC(A + 2ω) POINT

X3406 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, 182 }: Central Line L(76, 182) { 4, 1691 }: Central Line L(4, 1691) { 3, 1916 }: Central Line L(3, 1916) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3095 = INTERSECTION OF LINES X(3)X(6) AND X(5)X(76) (0.6556, 0.3590) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1691 = RADICAL TRACE OF CIRCUMCIRCLE AND 1st LEMOINE CIRCLE (0.9991, 0.9263) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1916 = TRILINEAR POLE OF PU(11) (1.0447, 0.0288) X_3406 = SEC(A + 2ω) POINT (0.4072, 0.1145) A B C X3095 X76 X182 X4 X1691 X3 X1916 X3406 (a, b, c) = (5, 6, 7) | AB = 1

X3406 (SEC(A + 2ω) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sec(A + 2\omega ) : :\)
Search Value (6, 9, 13): \(3.149957481949\)

X3406 lies on the Kiepert hyperbola.

▸ Incident Central Lines3 lines

{3, 1916}, {4, 1691}, {76, 182}

▸ Functional Relations2 relations

Notes

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X3406. (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = 2ω. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X3406. (Randy Hutson, July 20, 2016)