X₁₈₈

2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE

X₁₈₈ = anticomplement of X₁₇₈

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

X₁₈₈ (2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 178}, {9, 173}

• Related Centers: X₂₆₆ (ISOGONAL CONJUGATE OF X₁₈₈), X₂ (CENTROID), X₁₇₈ (2nd MID-ARC POINT), X₉ (MITTENPUNKT), X₁₇₃ (CONGRUENT ISOSCELIZERS POINT)

Trilinears: \(\csc A/2 : \csc B/2 : \csc C/2\)

Barycentrics: 1/(sin(B/2) sin(C/2) + sin(A/2)) : : c.f., X₅₄₅₁

Search Value (6, 9, 13): \(3.169616971348\)

▸ Alternate Coordinates6 forms

• Trilinears: \([bc(b + c - a)]^{1/2} : :\)

• Trilinears: \(\csc A \cos A/2 : :\)

• Trilinears: \(|AX(1)| : |BX(1)| : |CX(1)|\)

• Trilinears: \(\sec A' : \sec B', \sec C'\), where J_aJ_bJ_c = excentral triangle

• Barycentrics: 1/(csc(B/2) csc(C/2) + csc(A/2)) : : c.f., X₅₄₅₁

• Barycentrics: \(\cos A/2 : \cos B/2 : \cos C/2\)

▸ Incident Central Lines6 lines

{1, 361}, {2, 178}, {9, 173}, {40, 164}, {166, 167}, {174, 266}

▸ Functional Relations15 relations

• X₁₈₈ = isogonal conjugate of X₂₆₆

• X₁₈₈ = isotomic conjugate of X₄₁₄₆

• X₁₈₈ = anticomplement of X₁₇₈

• X₁₈₈ = X₂-Ceva conjugate of X₂₃₆

• X₁₈₈ = cevapoint of X₁ and X₁₆₄

• X₁₈₈ = X₂₅₉-cross conjugate of X₁₇₄

• X₁₈₈ = crosssum of X₁ and X₃₆₁

• X₁₈₈ = X₁₈₈-beth conjugate of X₂₆₆

• X₁₈₈ = SS(A→J_a) of X₄, where J_aJ_bJ_c is the excentral triangle

• X₁₈₈ = isotomic conjugate of X₄₁₄₆

• X₁₈₈ = X₆₅-of-excentral-triangle

• X₁₈₈ = perspector of circumconic centered at X₂₃₆

• X₁₈₈ = center of circumconic that is locus of trilinear poles of lines passing through X₂₃₆

• X₁₈₈ = excentral-to-ABC trilinear image of X₁

• X₁₈₈ = excentral-to-ABC barycentric image of X₁₆₄

Notes

Let J_aJ_bJ_c be the excentral triangle of ABC, so that J_a = \(-1 : 1 : 1\) (trilinears). Let A″ be the point where the bisector of angle BJ_aC meets the line BC. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₈₈. (Seiichi Kirikami, February 14, 2010)

Let E_a be the ellipse with B and C as foci and passing through the A-excenter, and define E_b and E_c cyclically. Let L_a be the line tangent to E_a at the A-excenter, and define L_b and L_c cyclically. Let A' = L_a∩BC, B' = L_b∩CA, C' = L_c∩AB. Then A', B', C' are collinear, and the trilinear pole of line A'B'C' = X₁₈₈. Note: The triangle formed by L_a, L_b, L_c is also the excentral triangle of the excentral triangle. Alternately, let A″ be the trilinear pole of line L_a, and define B″, C″ cyclically. The lines AA″, BB″ and CC″ concur at X₁₈₈; see also X₁₇₄. (Randy Hutson, December 2, 2017)