X₁₉₈₆

HATZIPOLAKIS REFLECTION POINT

Let H_aH_bH_c be the orthic triangle of triangle ABC. Let A_B be the reflection of A in H_c, and define A_C, B_C, B_A, C_A, C_B functionally. Then the nine-point circles of the triangles

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

X₁₉₈₆ (HATZIPOLAKIS REFLECTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 74}, {4, 94}

• Related Centers: X₄ (ORTHOCENTER), X₄₀₃ (OF THE ORTHIC TRIANGLE), X₆ (SYMMEDIAN POINT), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₉₄ (ISOGONAL CONJUGATE OF X₅₀)

Trilinears: \((1 + \cos 2B + \cos 2C) \sin 3A \csc 2A : :\)

Search Value (6, 9, 13): \(2.548224457367\)

▸ Incident Central Lines10 lines

{4, 94}, {6, 74}, {24, 110}, {25, 399}, {113, 403}, {125, 389}, {186, 323}, {542, 1843}, {648, 1300}, {1844, 1845}

▸ Functional Relations9 relations

• X₁₉₈₆ = reflection of X_i in X_j for these (i,j): (4, 1112), (74, 974), (125, 389)

• X₁₉₈₆ = X₄-Ceva conjugate of X₄₀₃

• X₁₉₈₆ = crosspoint of X₄ and X₁₈₆

• X₁₉₈₆ = crosssum of X₃ and X₂₆₅

• X₁₉₈₆ = X₈₀-of-orthic-triangle if ABC is acute

• X₁₉₈₆ = antigonal conjugate of X₄ wrt orthic triangle

• X₁₉₈₆ = antipode of X₄ in Hatzipolakis-Lozada hyperbola

• X₁₉₈₆ = perspector of orthic triangle and Hatzipolakis-Moses triangle

• X₁₉₈₆ = X₁₁-of-circumorthic-triangle if ABC is acute

Notes

AA_BA_C, BB_CB_A, CC_AC_B,

concur in X₁₉₈₆. (Antreas Hatzipolakis, Hyacinthos 7868, 9/12/03; coordinates by Barry Wolk, Hyacinthos 7876, 9/13/03)

Let A'B'C' = cevian triangle of X₁₈₆. Let A″, B″, C″ be the inverse-in-circumcircle of A', B', C'. The lines AA″, BB″, CC″ concur in X₁₉₈₆. (Randy Hutson, December 2, 2017)

Let H_aH_bH_c = orthic triangle. Let H_bH_cA″ be the triangle similar to ABC such that segment H_aA″ crosses the line H_bH_c. Define B″ and C″ cyclically. Equivalently, A″ is the reflection of A in H_bH_c, and cyclically for B″ and C″. Equivalently, A″ is the isogonal conjugate of H_a wrt AH_bH_c, and cyclically for B″ and C″. The lines H_aA″, H_bB″, H_cC″ concur in X₁₉₈₆. (Randy Hutson, December 2, 2017)

Let H_a be the foot of the A-altitude. Let B_a, C_a be the feet of perpendiculars from H_a to CA, AB, resp. Let N_a be the nine-point center of H_aB_aC_a. Define N_b and N_c cyclically. The lines H_aN_a, H_bN_b, H_cN_c concur in X₁₉₈₆. (Randy Hutson, December 2, 2017)

Let H_aH_bH_c be the orthic triangle. Let O_a be the A-Johnson circle of triangle AH_bH_c, and define O_b and O_c cyclically. The circles O_a, O_b, O_c concur in X₁₉₈₆. (Randy Hutson, July 31 2018)