X₁₃₈₄

2nd GRINBERG HOMOTHETIC CENTER

Let G_A be the circumcenter of triangle BCG, where G - centroid(ABC). Define G_B and G_C cyclically. Triangle G_AG_BG_C is homothetic to the pedal triangle of X₆, and X₁₃₈₄ is the center of the homothety.

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

X₁₃₈₄ (2nd GRINBERG HOMOTHETIC CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {25, 111}

• Related Centers: X₁₃₈₃ (1st GRINBERG HOMOTHETIC CENTER), X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₁₁₁ (PARRY POINT)

Trilinears: \(a(b^{2} + c^{2} - 5a^{2}) : :\)

Search Value (6, 9, 13): \(-0.145400755392\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sin(A + \omega ) - 5 \sin(A - \omega ) : :\)

▸ Incident Central Lines5 lines

{3, 6}, {25, 111}, {55, 609}, {230, 381}, {385, 1003}

▸ Functional Relations5 relations

• X₁₃₈₄ = X₁₃₈₃-Ceva conjugate of X₆

• X₁₃₈₄ = isogonal conjugate of X₅₄₈₅

• X₁₃₈₄ = intersection of tangents at PU(2) to hyperbola {X₆,PU(1),PU(2)}

• X₁₃₈₄ = {X₃,X₆}-harmonic conjugate of X₅₀₂₄

• X₁₃₈₄ = inverse-in-1st-Brocard-circle of X₅₀₂₄

Notes

Let DEF be the circumsymmedial triangle and let (O_a) be the circle tangent at D to circumcircle and with center O_a=AH∩ OD. Define (O_b) and (O_c) cyclically. The radical center of the circles (O_a), (O_b), (O_c) is X₁₃₈₄. (Angel Montesdeoca, October 2, 2018)