X₁₃₃₈

2nd WERNAU POINT

Let A'BC be the internal equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X₁₃₃₈. For details, see X₁₃₃₇.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₃₃₈ (2nd WERNAU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 617}, {1157, 1337}

• Related Centers: X₃₄₈₀ (ISOGONAL CONJUGATE OF X₁₃₃₈), X₄ (ORTHOCENTER), X₆₁₇ (ANTICOMPLEMENT OF X₁₄), X₁₁₅₇ (INVERSE-IN-CIRCUMCIRCLE OF X₅₄), X₁₃₃₇ (1st WERNAU POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[x_{1} - 2Dx_{2}/\sqrt{3}]\), where \(D = \operatorname{area}(ABC)\), \(x_{1}\), \(x_{2}\) are as at X₁₃₃₇.

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}(2DU + 3^{1/2}VW)/(4D - 3^{1/2}a^{2})\), D, U, V, W as at X₁₃₃₇; see Hyacinthos #8874

Search Value (6, 9, 13): \(-6.939133376623\)

▸ Alternate Coordinates1 form

• Barycentrics: a^2*(2*(-a^2+b^2+c^2)*S+\(\sqrt{3}\)*(a^2-c^2+b^2)*(a^2-b^2+c^2))*(\(\sqrt{3}\)*b^2-2*S)*(\(\sqrt{3}\)*c^2-2*S) : : (César Lozada, December 15, 2019)

X₁₃₃₈ lies on the Neuberg cubic.

▸ Incident Central Lines4 lines

{4, 617}, {16, 3458}, {399, 3440}, {1157, 1337}

▸ Functional Relations4 relations

• X₁₃₃₈ = isogonal conjugate of X₃₄₈₀

• X₁₃₃₈ = anticomplement of X₃₃₄₉₈

• X₁₃₃₈ = antigonal conjugate of X₆₆₁

• X₁₃₃₈ = symgonal of X₁₅

Notes

X₁₃₃₈ is the tangential of X₁₆ on the Neuberg cubic.