X₄₄₂

COMPLEMENT OF SCHIFFLER POINT

X₄₄₂ is the complement of Schiffler point.

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

X₄₄₂ (COMPLEMENT OF SCHIFFLER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {79, 191}, {2, 3} (Euler line)

• Related Centers: X₇₉ (ISOGONAL CONJUGATE OF X₃₅), X₁₉₁ (-CEVA CONJUGATE OF X₁), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(bc(v + w) : ca(w + u) : ab(u + v)\), where \(u : v : w\) = X₂₈₄; e.g., \(u(A,B,C) = (\sin A)/(\cos B + \cos C)\)

Barycentrics: \(v + w : w + u : u + v\)

Search Value (6, 9, 13): \(2.355469758576\)

▸ Incident Central Lines11 lines

{2, 3}, {8, 495}, {9, 46}, {10, 12}, {11, 214}, {100, 943}, {115, 120}, {119, 125}, {274, 325}, {388, 956}, {392, 946}

▸ Functional Relations12 relations

• X₄₄₂ = midpoint of X₇₉ and X₁₉₁

• X₄₄₂ = isogonal conjugate of X₁₁₇₅

• X₄₄₂ = inverse-in-orthocentroidal-circle of X₄₀₅

• X₄₄₂ = complement of X₂₁

• X₄₄₂ = complementary conjugate of X₉₆₀

• X₄₄₂ = X₁₀₀-Ceva conjugate of X₅₂₃

• X₄₄₂ = crosspoint of X₂₆₄ and X₃₂₁

• X₄₄₂ = crosssum of X₁₈₄ and X₁₃₃₃

• X₄₄₂ = perspector of Feuerbach triangle and 2nd extouch triangle

• X₄₄₂ = homothetic center of 4th Euler triangle and 2nd extouch triangle

• X₄₄₂ = X₅₄-of-2nd-extouch-triangle

• X₄₄₂ = X₉₇₃-of-excentral-triangle

Notes

As a point on the Euler line, X₄₄₂ has Shinagawa coefficients \((\$aS_{A}\$ + abc,\$aS_{A}\$ - abc)\)

Let I_A, I_B, I_C be the excenters, let A_B, A_C be the projections of A onto I_AI_B and I_AI_C, respectively, and define B_C, B_A and C_A, C_B cyclically. The Euler lines of the four triangles ABC, AA_BA_C, BB_CB_A, CC_AC_B concur in X₄₄₂. (Jean-Pierre Ehrmann, 11/24/01)

In the plane of triangle ABC, let DEF denote the intouch triangle of ABC, and let

• H_A = orthocenter of IBC

• M_A = midpoint of segment BC

• N_A = midpoint of the arc BC which does not include A

• \(S_{A}\) = reflection of I in line DE,

• and define cyclically the points H_B, H_C, M_B, M_C, N_B, N_C, and \(S_{B}\), \(S_{C}\). The lines \(S_{A}\)M_A, \(S_{B}\)M_B, \(S_{CA}\)M_C concur in X₄₄₂, and X₄₄₂ is the pole, with respect to the incircle, of the perspectrix of the triangles H_AH_BH_C and N_AN_BN_C. (Dominik Burek, January 18 2012)

Let J_aJ_bJ_c be the excentral triangle. X₄₄₂ is the radical center of the nine-point circles of J_aBC, J_bCA, J_cAB. (Randy Hutson, September 14, 2016)

Let (J_a) be the A-excircle of the A-altimedial triangle, and define (J_b) and (J_c) cyclically. X₄₄₂ is the radical center of (J_a), (J_b), (J_c). (Randy Hutson, November 2, 2017)