X₈₄₉

4th HATZIPOLAKIS-YIU POINT

Let D denote the circumcircle of triangle ABC. Let D_A be the circle tangent to sideline BC and tangent to D at A. Let B_A = AC∩D_A and C_A = AB∩D_A, and define C_B, C_B and A_C, A_B cyclically. Define A' = C_BA_B∩A_CB_C, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to ABC, and X₈₄₉ is the center of homothety. See A. Hatzipolakis and P. Yiu, Hyacinthos #2056-2070, December, 2000.

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

X₈₄₉ (4th HATZIPOLAKIS-YIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {36, 58}, {32, 163}

• Related Centers: X₂₄₉ (ISOGONAL CONJUGATE OF X₁₁₅), X₁₆₃ (TRILINEAR PRODUCT X₆*X₁₁₀), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₅₈ (ISOGONAL CONJUGATE OF X₁₀), X₃₂ (3rd POWER POINT)

Trilinears: \([a/(b + c)]^{2} : [b/(c + a)]^{2} : [c/(a + b)]^{2}\)

Barycentrics: \(a[a/(b + c)]^{2} : b[b/(c + a)]^{2} : c[c/(a + b)]^{2}\)

Search Value (6, 9, 13): \(0.287839619174\)

▸ Incident Central Lines6 lines

{32, 163}, {36, 58}, {110, 595}, {249, 1110}, {741, 827}, {757, 763}

▸ Functional Relations5 relations

• X₈₄₉ = isogonal conjugate of X₁₀₈₉

• X₈₄₉ = X₂₄₉-Ceva conjugate of X₁₆₃

• X₈₄₉ = crosspoint of X₅₈ and X₅₀₁

• X₈₄₉ = crosssum of X₁₀ and X₅₀₂

• X₈₄₉ = trilinear square of X₅₈