X₃₅₄

WEILL POINT

X₃₅₄ = crosspoint of X₁ and X₇

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

X₃₅₄ (WEILL POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 3681}

• Related Centers: X₂ (CENTROID), X₃₆₈₁ (com(EXTOUCH TRIANGLE))

Trilinears: \((b - c)^{2} - ab - ac : :\)

Barycentrics: \(a[(b - c)^{2} - ab - ac] : :\)

Search Value (6, 9, 13): \(1.303447777473\)

▸ Alternate Coordinates3 forms

• Trilinears: \(2 + \cos B + \cos C : :\)

• Trilinears: \(\cos^{2}(B/2) + \cos^{2}(C/2) : :\)

• Trilinears: \(b(\cot B/2) + c(\cot C/2) : :\)

▸ Incident Central Lines18 lines

{1, 3}, {2, 210}, {6, 374}, {7, 479}, {11, 118}, {37, 38}, {42, 244}, {44, 748}, {48, 584}, {63, 1001}, {81, 105}, {142, 3059}, {278, 955}, {373, 375}, {388, 938}, {392, 551}, {516, 553}, {1418, 2293}

▸ Functional Relations24 relations

• X₃₅₄ = isogonal conjugate of X₂₃₄₆

• X₃₅₄ = complement of X₃₆₈₁

• X₃₅₄ = anticomplement of X₃₇₄₀

• X₃₅₄ = inverse-in-incircle of X₁₁₅₅

• X₃₅₄ = reflection of X_i in X_j for these (i,j): (210, 2), (392, 551)

• X₃₅₄ = X₁₀₁-Ceva conjugate of X₅₁₃

• X₃₅₄ = crosspoint of X₁ and X₇

• X₃₅₄ = crosssum of X₁ and X₅₅

• X₃₅₄ = X₂-of-intouch-triangle

• X₃₅₄ = homothetic center of intouch triangle and inverse-in-incircle triangle

• X₃₅₄ = centroid of inverse-in-incircle triangle

• X₃₅₄ = pole of antiorthic axis wrt incircle

• X₃₅₄ = {X₁,X₄₀}-harmonic conjugate of X₃₃₀₃

• X₃₅₄ = {X₁,X₆₅}-harmonic conjugate of X₃₀₅₇

• X₃₅₄ = {X₃₅₁₃,X₃₅₁₄}-harmonic conjugate of X₆₅

• X₃₅₄ = centroid of the six intersections of the Conway circle and the sidelines of ABC

• X₃₅₄ = centroid of the six intersections of the sidelines of ABC and the antiparallels to sidelines through X₁

• X₃₅₄ = inverse-in-{circumcircle, incircle}-inverter of X₃₆

• X₃₅₄ = cevian isogonal conjugate of X₇ = intouch isogonal conjugate of X₇

• X₃₅₄ = bicentric sum of PU(94)

• X₃₅₄ = PU(94)-harmonic conjugate of X₆₅₀

• X₃₅₄ = X₃₈₁-of-incircle-circles-triangle

• X₃₅₄ = {X₁,X₅₇}-harmonic conjugate of X₅₅

• X₃₅₄ = centroid of the six intersections of the Adams’ circle and the sidelines of ABC

Notes

X₃₅₄ is the perspector of the intangents triangle and the triangle Q_AQ_BQ_C described at X₃₅₉₈. (Peter Moses, Nov. 4, 2010)

Let A', B', C' be the inverse-in-{circumcircle, incircle}-inverter of A, B, C. Let A″B″C″ be the tangential triangle of A'B'C'. Then A″B″C″ is perspective to ABC at X₃₅₄. (Randy Hutson, December 26, 2015)

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 16.

Let A_b be the intersection, other than B, of circle {X₁,B,C}} and line AB. Let A_c be the intersection, other than C, of circle {X₁,B,C}} and line AC. Define B_c, B_a, C_a, C_b cyclically. X₃₅₄ is the centroid of AbAcBcBaCaCb. Note that the lines A_bA_c, B_cB_a, C_aC_b bound the intangents triangle. (Randy Hutson, December 26, 2015)