X₄₉₄

2nd VAN LAMOEN HOMOTHETIC CENTER

X₄₉₄ is the homothetic center of triangle ABC and the Lucas(-1) homothetic triangle; see X₃₇₁.

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

X₄₉₄ (2nd VAN LAMOEN HOMOTHETIC CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {25, 372}, {39, 493}

• Related Centers: X₃₉₄ (-CEVA CONJUGATE OF X₃), X₄₉₃ (1st VAN LAMOEN HOMOTHETIC CENTER), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₃₇₂ ({X), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(1/(\sin A - \sin B \sin C) : :\)

Barycentrics: \((\sin A)(\sin A - \sin B \sin C) : :\)

Search Value (6, 9, 13): \(-40.639585709451\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{2}/(a^{2} - S) : :\)

▸ Incident Central Lines3 lines

{25, 372}, {39, 493}, {394, 1505}

▸ Functional Relations4 relations

• X₄₉₄ = isogonal conjugate of X₃₀₆₉

• X₄₉₄ = X₃₉₄-cross conjugate of X₄₉₃

• X₄₉₄ = perspector of ABC and unary cofactor triangle of inner Vecten triangle

• X₄₉₄ = perspector of ABC and unary cofactor triangle of Lucas antipodal triangle

Notes

X₄₉₄ is a homothetic center of triangles associated with squares that circumscribe ABC. For details and reference, see X₄₉₃. (Floor van Lamoen, 4/27/98)

Let A″ be the intersection of line BC and the common tangent to the B- and C-Lucas(-1) circles (at their touchpoint). Define B″ and C″ cyclically. Then A″B″C″ is the cevian triangle of X₄₉₄. Also, X₄₉₄ is the point of intersection of the tangents at X₃₇₂ and X₄₈₆ to the orthocubic K006. (Randy Hutson, July 23, 2015)