X₁₃₈₃

1st GRINBERG HOMOTHETIC CENTER

X₁₃₈₃ = vertex conjugate of X₂ and X₆

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

X₁₃₈₃ (1st GRINBERG HOMOTHETIC CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 23}, {32, 111}

• Related Centers: X₅₉₉ (ISOTOMIC CONJUGATE OF X₅₉₈), X₆ (SYMMEDIAN POINT), X₂₃ (FAR-OUT POINT), X₃₂ (3rd POWER POINT), X₁₁₁ (PARRY POINT)

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

Barycentrics: \(\operatorname{area}(A'BC) : :\), where Cm_aCm_bCm_c is the circummedial triangle

Search Value (6, 9, 13): \(0.209320409520\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\operatorname{area}(A'BC) : :\), where A'B'C' is the circumsymmedial triangle

• Barycentrics: \(\operatorname{area}(A'BC) : :\), where A'B'C' is the 4th Brocard triangle

▸ Incident Central Lines3 lines

{2, 187}, {6, 23}, {32, 111}

▸ Functional Relations4 relations

• X₁₃₈₃ = isogonal conjugate of X₅₉₉

• X₁₃₈₃ = isotomic conjugate of X₉₄₆₄

• X₁₃₈₃ = vertex conjugate of X₂ and X₆

• X₁₃₈₃ = cevapoint of X₆ and X₁₃₈₄

Notes

Let A'B'C' be the circumcevian triangle of X₂, and let P(A) be the line through A' parallel to line BC. Define P(B) and P(C) cyclically. Let A″ = P(B)∩P(C), and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC, and X₁₃₈₃ is the center of the homothety.

The center of the homothety is also X₁₃₈₃ if “circumcevian triangle of X₆ (circumsymmedial triangle)” or “4th Brocard triangle” is substituted for “circumcevian triangle of X₂”. (Randy Hutson, March 21, 2019)