X₁₇₄

YFF CENTER OF CONGRUENCE

In notes dated 1987, Yff raises this question concerning certain triangles lying within ABC: can isoscelizers (defined at X₁₇₃), P_BQ_C, P_CQ_A, P_AQ_B, be constructed so that, on putting

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

X₁₇₄ (YFF CENTER OF CONGRUENCE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 2090}

• Related Centers: X₂ (CENTROID), X₂₀₉₀ (1st STEVANOVIC PERSPECTOR)

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

Barycentrics: \(\sin A/2 : \sin B/2 : \sin C/2\)

Search Value (6, 9, 13): \(1.175132454771\)

▸ Alternate Coordinates4 forms

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

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

• Trilinears: \(|AA'| : |BB'| : |CC'|\), where J_aJ_bJ_c = excentral triangle

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

▸ Incident Central Lines9 lines

{1, 167}, {2, 236}, {7, 234}, {57, 173}, {175, 483}, {176, 1143}, {188, 266}, {481, 1127}, {558, 1489}

▸ Functional Relations18 relations

• X₁₇₄ = isogonal conjugate of X₂₅₉

• X₁₇₄ = anticomplement of X₂₀₉₀

• X₁₇₄ = X₅₀₈-Ceva conjugate of X₁₈₈

• X₁₇₄ = cevapoint of X_i and X_j for these (i,j): (1, 173), (259, 266)

• X₁₇₄ = X_i-cross conjugate of X_j for these (i,j): (1, 1488), (177, 7), (259, 188)

• X₁₇₄ = crosssum of X₁ and X₅₀₃

• X₁₇₄ = X₅₅₆-beth conjugate of X₅₅₆

• X₁₇₄ = SS(A→J_a)-of-X₂, where J_aJ_bJ_c is the excentral triangle

• X₁₇₄ = SS(A→J_a)-of-X₂₂₆, where J_aJ_bJ_c is the excentral triangle

• X₁₇₄ = isotomic conjugate of X₅₅₆

• X₁₇₄ = X₅₅-of-intouch triangle

• X₁₇₄ = X₅₅-of-Yff central triangle

• X₁₇₄ = homothetic center of intouch triangle and Yff central triangle

• X₁₇₄ = homothetic center of ABC and the intangents triangle of the intouch triangle.

• X₁₇₄ = {X₈₁₃₄,X₈₁₃₆}-harmonic conjugate of X₈₁₂₃

• X₁₇₄ = {X₈₁₃₇,X₈₁₃₉}-harmonic conjugate of X₈₁₂₄

• X₁₇₄ = X₁₈₂₄-of-excentral-triangle

• X₁₇₄ = excentral-to-ABC trilinear image of X₁₆₅

Notes

Let E_a be the ellipse with B and C as foci and passing through X₁, and define E_b and E_c cyclically. Let L_a be the line tangent to E_a at X₁, 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 X₁₇₄ = trilinear pole of line A'B'C'. The line A'B'C' meets the line at infinity at the isogonal conjugate of X₃₆₅₉. Alternately, let A″ be the trilinear pole of line L_a, and define B″ and C″ cyclically. The lines AA″, BB″ and CC″ concur at X₁₇₄; see also X₁₈₈. The points A″, B″, C″ lie on the circumconic centered at X₉. (Randy Hutson, December 10, 2016)

Let J_aJ_bJ_c be the excentral triangle. X₁₇₄ is the trilinear pole of the Monge line of the incircles of BCJ_a, CAJ_b, ABJ_c. (Randy Hutson, December 10, 2016)

\(R_{A}\) = P_AQ_B∩P_BQ_C, \(R_{B}\) = P_BQ_C∩P_CQ_A, \(R_{C}\) = P_CQ_A∩P_AQ_B,

the following four triangles are congruent:

P_AQ_A\(R_{A}\), P_BQ_B\(R_{B}\), P_CQ_C\(R_{C}\), \(R_{A}\)\(R_{B}\)\(R_{C}\) ?

After proving that the answer is yes, Yff moves the three isoscelizers in such a way that the three outer triangles, stay congruent and the inner triangle (called the Yff central triangle), \(R_{A}\)\(R_{B}\)\(R_{C}\), shrinks to X₁₇₄.

Let D be the point on side BC such that (angle BID) = (angle DIC), and likewise for point E on side CA and point F on side AB. The lines AD, BE, CF concur in X₁₇₄. (Seiichi Kirikami, Jan. 29, 2010)

Generalization: if I is replaced by an arbitrary point P =\(p : q : r\) (trilinears), then the lines AD, BE, CF concur in the point K(P) =\(f(p,q,r,A) : f(q,r,p,B) : f(r,p,q,C)\), where \(f(p,q,r,A) = (q^{2} + r^{2} + 2qr \cos A)^{-1/2}\). Moreover, if P* is the inverse of P in the circumcircle, then K(P*) = K(P). (Peter Moses, Feb. 1, 2010, based on Seiichi Kirikami’s construction of X₁₇₄)

X₁₇₄ is the homothetic center of ABC and the extangents triangle of the intouch triangle. (Randy Hutson, 9/23/2011)