X₁₇₃

CONGRUENT ISOSCELIZERS POINT

Let P_B on sideline AC and Q_C be equidistant from A, so that AP_BQ_C is an isosceles triangle. The line P_BQ_C is called an isoscelizer. The lines P_BQ_C, P_CQ_A, P_AQ_B concur in X₁₇₃. (P. Yff, unpublished notes, 1989)

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

X₁₇₃ (CONGRUENT ISOSCELIZERS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 168}, {9, 177}

• Related Centers: X₂₅₈ (CONGRUENT INCIRCLES ISOSCELIZER POINT), X₁ (INCENTER), X₁₆₈ (MITTENPUNKT OF EXCENTRAL TRIANGLE), X₉ (MITTENPUNKT), X₁₇₇ (1st MID-ARC POINT)

Trilinears: \(\cos B/2 + \cos C/2 - \cos A/2 : :\)

Search Value (6, 9, 13): \(0.865451887564\)

▸ Alternate Coordinates6 forms

• Trilinears: \(\tan A/2 + \sec A/2 : \tan B/2 + \sec B/2 : \tan C/2 + \sec C/2\) (M. Iliev, 4/12/07)

• Trilinears: \(b(\csc B/2) + c(\csc C/2) - a(\csc A/2) : :\)

• Trilinears: \(\cot A'/2 : :\), where J_aJ_bJ_c = excentral triangle

• Trilinears: \(b' + c' - a' : :\), where a’, b’, c’ are sidelengths of excentral triangle

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

• Trilinears: [distance from A to far side of A-excircle] : :

▸ Incident Central Lines7 lines

{1, 168}, {9, 177}, {57, 174}, {164, 504}, {180, 483}, {503, 844}, {505, 1130}

▸ Functional Relations11 relations

• X₁₇₃ = isogonal conjugate of X₂₅₈

• X₁₇₃ = anticomplement of X₃₄₈₄₉

• X₁₇₃ = X₁₇₄-Ceva conjugate of X₁

• X₁₇₃ = SS(A→J_a)-of-X₉, where J_aJ_bJ_c is the excentral triangle

• X₁₇₃ = X₁₉-of-excentral triangle

• X₁₇₃ = X₁₉-of-Yff central triangle

• X₁₇₃ = homothetic center of excentral triangle and Yff central triangle

• X₁₇₃ = homothetic center of ABC and orthic triangle of Yff central triangle

• X₁₇₃ = homothetic center of ABC and extangents triangle of excentral triangle

• X₁₇₃ = excentral isogonal conjugate of X₈₄₅

• X₁₇₃ = X_i-aleph conjugate of X_j for these (i,j): (1, 503), (2, 504), (174, 173), (188, 845), (366, 164), (507, 1), (508, 362), (509, 361)

Notes

The intouch triangle of the intouch triangle of triangle ABC is perspective to triangle ABC, and X₁₇₃ is the perspector. (Eric Danneels, Hyacinthos 7892, 9/13/03)

Also, X₁₇₃ = X₁₄₈₆-of-the-intouch-triangle. (Darij Grinberg; see notes at X₁₄₈₅ and X₁₄₈₆.)