X3399

SEC(A - 2ω) POINT

X3399 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) { 262, 1506 }: Central Line L(262, 1506) { 5, 1916 }: Central Line L(5, 1916) { 597, 1576 }: Central Line L(597, 1576) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3102 = INNER VERTEX OF THE BROCARD SQUARE (0.5362, 0.1619) X_3103 = OUTER VERTEX OF THE BROCARD SQUARE (0.6194, 0.2992) X_262 = ISOGONAL CONJUGATE OF X(182) (0.5893, 0.2856) X_1506 = INSIMILICENTER OF MOSES AND NINE-POINT CIRCLES (0.5676, 0.2392) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_1916 = TRILINEAR POLE OF PU(11) (1.0447, 0.0288) X_597 = MIDPOINT OF X(2) AND X(6) (0.5687, 0.2336) X_1576 = ISOGONAL CONJUGATE OF X(850) (0.3750, 0.6166) X_3399 = SEC(A - 2ω) POINT (0.5694, 0.2429) A B C X3102 X3103 X262 X1506 X5 X1916 X597 X1576 X3399 (a, b, c) = (5, 6, 7) | AB = 1

X3399 (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): \(0.439523311932\)

X3399 lies on the Kiepert hyperbola.

▸ Incident Central Lines3 lines

{5, 1916}, {262, 1506}, {597, 1576}

▸ Functional Relations2 relations

Notes

Let A'B'C' be the 1st Brocard triangle. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X3399. (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 = 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 X3399. (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed outward 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 X3399. (Randy Hutson, July 20, 2016)