X₃₉₉

PARRY REFLECTION POINT

Let L, M, N be lines through A, B, C, respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, let M' be the reflection of M in sideline CA, and let N' be the reflection of N in sideline AB. The lines L', M', N' concur in X₃₉₉, as proved in

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

X₃₉₉ (PARRY REFLECTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 13}, {3, 74}

• Related Centers: X₁₁₃₈ (ISOGONAL CONJUGATE OF X₃₉₉), X₆ (SYMMEDIAN POINT), X₁₃ (1st ISOGONIC CENTER), X₃ (CIRCUMCENTER), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT)

Trilinears: \(5 \cos A - 4 \cos B \cos C - 8 \sin B \sin C \cos^{2} A : :\)

Barycentrics: \(Barycentrics a^{2}[a^{8} - 4a^{6}(b^{2} + c^{2}) + a^{4}(6b^{4} + b^{2}c^{2} + 6c^{4}) - a^{2}(4b^{6} - b^{4}c^{2} - b^{2}c^{4} + 4 c^{6}) + (b^{2} - c^{2})^{2}(b^{4} + 4b^{2}c^{2} + c^{4})] : :\)

Search Value (6, 9, 13): \(-4.967236945993\)

X₃₉₉ lies on the Neuberg cubic.

▸ Incident Central Lines10 lines

{1, 3065}, {3, 74}, {4, 195}, {6, 13}, {30, 146}, {155, 382}, {394, 541}, {1337, 3441}, {1338, 3440}, {3466, 3483}

▸ Functional Relations12 relations

• X₃₉₉ = isogonal conjugate of X₁₁₃₈

• X₃₉₉ = reflection of X_i in X_j for these (i,j): (3, 110), (74, 1511), (265, 113)

• X₃₉₉ = X_i-Ceva conjugate of X_j for these (i,j): (30, 3), (323, 6)

• X₃₉₉ = inverse-in-circumcircle of X₁₅₁₁

• X₃₉₉ = tangential isogonal conjugate of X₂₀₇₀

• X₃₉₉ = antigonal image of X₁₁₁₇

• X₃₉₉ = orthocentroidal-to-ABC similarity image of X₃

• X₃₉₉ = 4th-Brocard-to-circumsymmedial similarity image of X₃

• X₃₉₉ = X₈₀-of-tangential-triangle if ABC is acute

• X₃₉₉ = orthologic center of these triangles: reflection to 1st Hyacinth

• X₃₉₉ = orthologic center of these triangles: reflection to AOA

• X₃₉₉ = perspector of circlecevian triangles of X₁₅ and X₁₆

Notes

Cyril Parry, Problem 10637, American Mathematical Monthly 105 (1998) 68.

In Cosmin Pohoata, “On the Parry reflection point,” :cite:`Forum Geometricorum <forum-geometricorum>` 8 (2008), 43-48 here, the following is proved:

Let Ref_a be the reflection of vertex A in line BC, and define Ref_b, Ref_c cyclically. Let A_tB_tC_t be the tangential triangle of ABC. The circumcircles of the triangles A_tRef_bRef_c, Ref_aB_tRef_c, Ref_aRef_bC_t concur in X₃₉₉. Moreover, the circumcircles of triangles Ref_aB_tC_t, A_tRef_bC_t, A_tB_tRef_c concur in a point Q = X8157), here named the Parry-Pohoata point. Barycentric coordinates for Q, of degree 22 in a,b,c, were found by J. F. Garcia Captitán (Hyacinthos #15827, Nov. 19, 2007) and are included in Pohoata’s article. Pohoata notes that the point Q lies on the circumcircle of the points X₃, X₄, X₃₉₉.

Let I, I_A, I_B, I_C, denote the incenter and excenters of ABC. Lawrence Evans (Hyacinthos #6878) found that the circumcircles of the triangles IA'I_A, IB'I_B, IC'_C concur in X₃₉₉.

The Pohoata article includes a proof that the circumcircles of the triangles A'I_BI_C, I_AB'I_C, I_AI_BC' also pass through X₃₉₉. Similar results involving the Fermat points, X₁₃ and X₁₄, are proved.

Pohoata reports that the following points are concyclic: X₁₃, X₁₆, X₁₁₀, X₃₉₉, X₁₃₃₈, as are the points X₁₄, X₁₅, X₁₁₀, X₃₉₉, X₁₃₃₇. Randy Hutson adds (Aug. 13, 2012) that the first of these circles also passes through X₂₃₈₁, and the second, through X₂₃₈₀.

X₃₉₉ is the point P on the line X₃X₇₄ for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, August 26, 2014)

Let A_aB_aC_a, A_bB_bC_b, A_cB_cC_c be the A-, B- and C-anti-altimedial triangles. Let L_a be the line through A_a parallel to the Euler line of A_aB_aC_a. Define L_b and L_c cyclically. The lines L_a, L_b, L_c concur in X₃₉₉. (Randy Hutson, November 2, 2017)

Let O_A be the circle centered at the A-vertex of the Walsmith triangle and passing through A; define O_B and O_C cyclically. X₃₉₉ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)