X₃₈₉

CENTER OF THE TAYLOR CIRCLE

X₃₈₉ = crosspoint of X₄ and X₅₄

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

X₃₈₉ (CENTER OF THE TAYLOR CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1216, 140}

• Related Centers: X₁₂₁₆ (ISOGONAL CONJUGATE OF X₁₁₇₉), X₁₄₀ (MIDPOINT OF X₃ AND X₅)

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

Barycentrics: \(a[\cos A - \cos 2A \cos(B - C)] : b[\cos B - \cos 2B \cos(C - A)] : c[\cos C - \cos 2C \cos(A - B)] : :\)

Search Value (6, 9, 13): \(3.019248079525\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\cos A - \sin A (\tan A + \tan B + \tan C) : :\)

• Trilinears: \(\cos A (\cot A \cot B \cot C) - \sin A : :\)

• Barycentrics: \(a^{2}[a^{6}(b^{2} + c^{2}) - 3a^{4}(b^{4} + c^{4}) + 3a^{2}(b^{2} - c^{2})^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}(b^{4} + c^{4})] : :\)

▸ Incident Central Lines8 lines

{3, 6}, {4, 51}, {24, 184}, {30, 143}, {54, 186}, {115, 129}, {217, 232}, {517, 950}

▸ Functional Relations19 relations

• X₃₈₉ = midpoint of X_i and X_j for these (i,j): (3, 52), (4, 185), (974, 1112)

• X₃₈₉ = reflection of X₁₂₁₆ in X₁₄₀

• X₃₈₉ = complement of X₅₅₆₂

• X₃₈₉ = anticomplement of X₁₁₇₉₃

• X₃₈₉ = polar-circle-inverse of X₆₇₆₁

• X₃₈₉ = Schoute-circle-inverse of X₂₂₀₅₂

• X₃₈₉ = circle-{X₃₇₁,X₃₇₂,PU(1),PU(39)}}-inverse of X₅₇₇

• X₃₈₉ = {X₁₅,X₁₆}-harmonic conjugate of X₂₂₀₅₂

• X₃₈₉ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₅₇₇

• X₃₈₉ = inverse-in-Brocard-circle of X₅₇₈

• X₃₈₉ = crosspoint of X₄ and X₅₄

• X₃₈₉ = crosssum of X_i and X_j for these (i,j): (3, 5), (6, 418)

• X₃₈₉ = orthology center of half-altitude and orthic triangles

• X₃₈₉ = complement of X₄ wrt orthic triangle

• X₃₈₉ = X₂₀-of-polar-triangle-of-complement-of-polar-circle

• X₃₈₉ = X₁₀-of-orthic-triangle if ABC is acute

• X₃₈₉ = X₄-of-1st-Hyacinth-triangle

• X₃₈₉ = excentral-to-ABC functional image of X₁₀

• X₃₈₉ = {X₆₁,X₆₂}-harmonic conjugate of X₃₂₈₄

Notes

If ABC is acute then X₃₈₉ is the Spieker center of the orthic triangle. Peter Yff reports (Sept. 19, 2001) that since X₃₈₉ is on the Brocard axis, there must exist T for which X₃₈₉ is \(\sin(A+T) : \sin(B+T) : \sin(C+T)\), and that tan(T) = - cot A cot B cot C.

Let H_A be the A-altitude of triangle ABC, and let A' be the midpoint of segment AH_A. Let L_A be the line through A' parallel to AO, where O denotes the circumcenter. Define L_B and L_C cyclically. The lines L_A, L_B, L_C concur in X₃₈₉. (Construction by Alexei Myakishev, March 24, 2010.)

Let O_A be the circle with center A tangent to line BC, and define O_B and O_C cyclically. X₃₈₉ is the radical center of the three circles. (Randy Hutson, 9/23/2011)

Let H_aH_bH_c be the orthic triangle, let A″ be the orthocenter of AH_bH_c, and define B″ and C″ cyclically. Triangle A″B″C″ is the medial triangle of the circumorthic triangle. The triangle A″B″C″ is homothetic to H_aH_bH_c, and the center of homothety is X₃₈₉. (Randy Hutson, 9/23/2011)

Let H_aH_bH_c be the orthic triangle. Let O_a be the circle with center A tangent to line H_bH_c, and define O_b and O_c cyclically. The radical circle of O_a, O_b, O_c is the Taylor circle, which is also the Spieker radical circle of the orthic triangle if ABC is acute. (Randy Hutson, December 2, 2017)

Let H_aH_bH_c be the orthic triangle. X₃₈₉ is the radical center of the polar circles of AH_bH_c, BH_cH_a, CH_aH_b. (Randy Hutson, December 2, 2017)

A construction of X₃₈₉ is given at 24162. (Antreas Hatzipolakis, August 29, 2016)

Let A'B'C' be the midheight triangle. Let A_B, A_C be the orthogonal projections of A' on CA, AB, resp. Define B_C, B_A, C_A, C_B cyclically. Let A″ = C_AA_C∩A_BB_A, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X₆. The lines A'A″, B'B″, C'C″ concur in X₃₈₉, which is also X₃-of-A″B″C″. (Randy Hutson, March 29, 2020)