X₂₂₁₇

X₂-ISOCONJUGATE OF X₅₇₃

X₂₂₁₇ = cevapoint of X₆₆₇ and X₂₁₇₀

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

X₂₂₁₇ (X₂-ISOCONJUGATE OF X₅₇₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 10}, {37, 48}

• Related Centers: X₆₆₇ (CROSSDIFFERENCE OF X₂ AND X₃₇), X₂₁₇₀ (-ISOCONJUGATE OF X₅₉), X₃ (CIRCUMCENTER), X₁₀ (SPIEKER CENTER), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID), X₄₈ (CROSSPOINT OF X₁ AND X₆₃)

Trilinears: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\), where f(a,b,c) is as in X₅₇₃

Barycentrics: \(a^{2}/f(a,b,c) : b^{2}/f(b,c,a): c^{2}/f(c,a,b)\)

Search Value (6, 9, 13): \(-2.449394554517\)

▸ Incident Central Lines12 lines

{1, 1437}, {3, 10}, {19, 1333}, {21, 1610}, {28, 158}, {37, 48}, {56, 225}, {58, 994}, {65, 603}, {75, 1444}, {1104, 1472}, {1953, 2214}

▸ Functional Relations4 relations

• X₂₂₁₇ = isogonal conjugate of X₃₈₆₉

• X₂₂₁₇ = cevapoint of X₆₆₇ and X₂₁₇₀

• X₂₂₁₇ = X₁₄₀₂-cross conjugate of X₆

• X₂₂₁₇ = X₂₁-vertex conjugate of X₂₁

Notes

Contribution by Angel Montesdeoa, September 10, 2017: (start)

In the plane of a triangle BAC, let

O = X₃ = circumcenter

U(O,q) = circle with center O and radius q

A'B'C' = excentral triangle

A*B*C* = polar triangle of A'B'C' with respect to U(O,q).

There exists a unique q such that the lines AA*, BB*, CC* concur, and the point of concurrence is X₂₂₁₇. Moreover, the radius and a barycentric equation for U(O,q) are given as follows:

q^2 = (4 r R^2 s(r+R))/(a b c) = (a b c (2 a b c+(a+b-c) (a-b+c)(-a+b+c)))/(2 (a+b-c) (a-b+c) (-a+b+c) (a+b+c));

abc(x^2+y^2+z^2) + 2a (a+b) (a+c)y z + 2b(b+c)(b+a)z x + 2c(c+a)(c+b)x y = 0. (end)