X₁₃₁₂

1st MOSES INTERSECTION

For a list of harmonic conjugates, click Tables at the top of this page.

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

X₁₃₁₂ (1st MOSES INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 1113}, {2, 3} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₁₁₁₃ (1st EULER-LINE-CIRCUMCIRCLE INTERSECTION), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(bcf(a,b,c) : caf(b,c,a): abf(c,a,b)\), where f(a,b,c) is the 1st barycentric given below

Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (J - 1)a^{2}S_{A} + 2(J + 1) S_{B} S_{C}\), where \(S_{A} = (b^{2} + c^{2} - a^{2})/2\), and \(S_{B}\) and \(S_{C}\) are defined cyclically.

Search Value (6, 9, 13): \(-2.642772044566\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (J - 1)\cos A + 2(J + 1)\cos B \cos C\), where \(J = |OH|/R\); see X₁₁₁₃

X₁₃₁₂ lies on the nine-point circle, the MacBeath inconic.

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations12 relations

• X₁₃₁₂ = 3X(2) - (3 + |OH|/R)*X₅ = 3(1 + |OH|/R)*X₂ - (3 + |OH|/R)*X₃

• X₁₃₁₂ = midpoint of X₄ and X₁₁₁₃

• X₁₃₁₂ = reflection of X₁₃₁₃ in X₅

• X₁₃₁₂ = complement of X₁₁₁₄

• X₁₃₁₂ = X₁₁₁₃-Ceva conjugate of X₅₂₃

• X₁₃₁₂ = inverse-in-polar-circle of X₁₁₁₄

• X₁₃₁₂ = excentral-to-ABC functional image of X₁₃₈₁

• X₁₃₁₂ = X₁₃₈₁-of-orthic-triangle if ABC is acute

• X₁₃₁₂ = {X₂,X₁₁₁₃}-harmonic conjugate of X₄₆₈

• X₁₃₁₂ = {X₂,X₈₅₈}-harmonic conjugate of X₁₃₁₃

• X₁₃₁₂ = {X₄,X₄₀₃}-harmonic conjugate of X₁₃₁₃

• X₁₃₁₂ = {X₄₂₇,X₄₆₈}-harmonic conjugate of X₁₃₁₃

Notes

As a point on the Euler line, X₁₃₁₂ has Shinagawa coefficients \((J - 1, J + 3)\).

X₁₃₁₂ is a point of intersection of the Euler line and the nine-point circle. Its antipode on the nine-point circle

is X₁₃₁₃. Of the two points, X₁₃₁₂ is the one closer to X₄. The asymptotes of the Jerabek hyperbola meet at X₁₂₅ on the nine-point circle. One of the asymptotes meets the circle again at X₁₃₁₂, and the other, at X₁₃₁₃. Thus, the points X₁₂₅, X₁₃₁₂, X₁₃₁₃ form a right triangle of which the midpoint of the hypotenuse X₁₃₁₂-to-X₁₃₁₃ is X₅. (Peter J. C. Moses, 3/14/2003)

Of the points other than X₁₂₅ in which the nine-point circle meets the asymptotes of the Jerabek hyperbola, X₁₃₁₂ is the one farther from X₃. (Randy Hutson, December 2, 2017)