X₁₄₉₈

REFLECTION OF X₆₄ IN X₃

Let (B_a, C_a) be the points where BC is cut by the parallels through A to the b- and c- altitudes, respectively, and define (C_b, A_b), (A_c, B_c) cyclically. Then the perpendicular bisectors of (B_a, C_a), (C_b, A_c) and (A_c, B_c) concur at X₁₄₉₈. (Hyacinthos #62, #72, #73)

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

X₁₄₉₈ (REFLECTION OF X₆₄ IN X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {64, 3}

• Related Centers: X₆₄ (ISOGONAL CONJUGATE OF X₂₀), X₃ (CIRCUMCENTER)

Trilinears: \((\sin A)(\tan^{2} B + \tan^{2} C - \tan^{2} A) : :\)

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

Search Value (6, 9, 13): \(-18.062598277647\)

▸ Alternate Coordinates1 form

• Trilinears: \(a [S^{2} - 2 (4 R^{2} - S_{A}) S_{A}] : :\)

X₁₄₉₈ lies on the Darboux cubic.

▸ Incident Central Lines15 lines

{1, 84}, {3, 64}, {4, 6}, {20, 394}, {24, 1192}, {25, 185}, {30, 155}, {40, 219}, {159, 1350}, {195, 382}, {1158, 1214}, {1490, 3347}, {2131, 3183}, {3182, 3354}, {3353, 3473}

▸ Functional Relations14 relations

• X₁₄₉₈ = reflection of X_i in X_j for these (i,j): (64, 3), (1350, 159)

• X₁₄₉₈ = isogonal conjugate of X₃₃₄₆

• X₁₄₉₈ = X_i-Ceva conjugate of X_j for these (i,j): (20, 3), (394, 6)

• X₁₄₉₈ = crosssum of X₁₂₂ and X₅₂₃

• X₁₄₉₈ = tangential isogonal conjugate of X₂₄

• X₁₄₉₈ = tangential isotomic conjugate of X(6

• X₁₄₉₈ = antipedal isogonal conjugate of X(64

• X₁₄₉₈ = Nagel point of tangential triangle if ABC is acute

• X₁₄₉₈ = perspector of antipedal triangle of X₆₄ and cevian triangle of X₂₀

• X₁₄₉₈ = intouch-to-ABC functional image of X₈

• X₁₄₉₈ = {X₁₂₉₆₄,X₁₂₉₇₀}-harmonic conjugate of X₆

• X₁₄₉₈ = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X₃), of X₆₄

• X₁₄₉₈ = perspector of ABC and the reflection in X₁₀₇₃ of the antipedal triangle of X₁₀₇₃

• X₁₄₉₈ = perspector of excentral triangle and cross-triangle of ABC and hexyl triangle

Notes

Let A'B'C' be the cevian triangle of X₁₀₃₂. Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₄₉₈. (Randy Hutson, December 10, 2016)

X₁₄₉₈ is the perspector of the tangential triangle and the reflection of triangle ABC in X₃; also, X₁₄₉₈ is X₈-of tangential triangle if ABC is acute. (Darij Grinberg, 6/2/03)

Let O_A be the circle centered at the A-vertex of the anticevian triangle of X₃ and passing through A; define O_B and O_C cyclically. X₁₄₉₈ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)