X₄₉₃

1st VAN LAMOEN HOMOTHETIC CENTER

X₄₉₃ is a homothetic center of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 4. Ingeschreven vierkanten (Floor van Lamoen, 4/27/98)

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

X₄₉₃ (1st 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, 371}, {39, 494}

• Related Centers: X₃₉₄ (-CEVA CONJUGATE OF X₃), X₄₉₄ (2nd VAN LAMOEN HOMOTHETIC CENTER), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₃₇₁ (KENMOTU POINT), 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): \(1.847641056142\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines3 lines

{25, 371}, {39, 494}, {394, 1504}

▸ Functional Relations4 relations

• X₄₉₃ = isogonal conjugate of X₃₀₆₈

• X₄₉₃ = X₃₉₄-cross conjugate of X₄₉₄

• X₄₉₃ = perspector of ABC and unary cofactor triangle of outer Vecten triangle

• X₄₉₃ = perspector of ABC and unary cofactor triangle of Lucas(-1) antipodal triangle

Notes

X₄₉₃ is the homothetic center of triangle ABC and the Lucas homothetic triangle; see X₃₇₁. Writing t for the ratio L:W at X₃₇₁, let L_A be the line through the intersections, other than A, of the A-Lucas(t) circle and sides CA and AB. Define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. Then the triangle A'B'C', here introduced as the Lucas(t) homothetic triangle, is homothetic to triangle ABC. If t = 1, the center of homothety is X₄₉₃; for t = -1, it is X₄₉₄; for t = 2, it is X₅₈₈; and for t = -2, it is X₅₈₉. (Randy Hutson, February 9, 2013)

Let A″ be the intersection of line BC and the common tangent to the B- and C-Lucas circles (at their touchpoint). Define B″ and C″ cyclically. Then A″B″C″ is the cevian triangle of X₄₉₃. Also, X₄₉₃ is the point of intersection of the tangents at X₃₇₁ and X₄₈₅ to the orthocubic K006. (Randy Hutson, July 23, 2015)