X3411

1st BROCARD-KIEPERT-FERMAT CUSP

X3411 and X3412 occur in Bernard Gibert’s study of the Brocard-Kiepert quartic, which has three cusps: X20 with Euler line as tangent; X3411 with line X5X13 as tangent, and X3412 with line X5X14 as tangent.
< 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) { 5, 13 }: Central Line L(5, 13) { 16, 20 }: Central Line L(16, 20) { 14, 382 }: Central Line L(14, 382) { 61, 631 }: Central Line L(61, 631) { 398, 548 }: Central Line L(398, 548) { 630, 3180 }: Central Line L(630, 3180) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7245, 0.6894) X_61 = ISOGONAL CONJUGATE OF X(17) (0.5764, 0.2283) X_631 = 3/5*OG (0.5224, 0.1608) X_398 = CROSSPOINT OF X(4) AND X(18) (0.5945, 0.2099) X_548 = MIDPOINT OF X(5) AND X(20) (0.4719, 0.0286) X_630 = COMPLEMENT OF X(18) (0.5844, 0.2341) X_3180 = -CEVA CONJUGATE OF X(2) (0.6674, 0.3397) X_3411 = 1st BROCARD-KIEPERT-FERMAT CUSP (0.7746, 0.4761) A B C X5 X13 X16 X20 X14 X382 X61 X631 X398 X548 X630 X3180 X3411 (a, b, c) = (5, 6, 7) | AB = 1

X3411 (1st BROCARD-KIEPERT-FERMAT CUSP) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos A - 2 \sin B \sin C - 3 \cos(A +\pi /3)\)
Barycentrics: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = 16S^{2} - 2 \cdot 3^{3/2}a^{2}S + a^{2}S_{A}\), where S = 2(area(ABC))
Search Value (6, 9, 13): \(-0.801182147994\)
▸ Alternate Coordinates2 forms

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \cos A + 4 \sin B \sin C - 3^{3/2}\sin A\)

  • Barycentrics: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = 20S^{2} - 2 \cdot 3^{3/2}S - S_{B}S_{C}\)

X3411 lies on the Brocard-Kiepert quartic, Q073.

▸ Incident Central Lines7 lines

{5, 13}, {6, 3412}, {14, 382}, {16, 20}, {61, 631}, {398, 548}, {630, 3180}

▸ Functional Relations1 relation