X₅₈₈

1st KENMOTU-VAN LAMOEN POINT

X₅₈₈ is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of X₃₇₁ and the points where the squares in the Kenmotu configuration with center X₃₇₁ meet sideline BC, and B' and C' are defined cyclically. See

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

X₅₈₈ (1st KENMOTU-VAN LAMOEN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {39, 589}

• Related Centers: X₆ (SYMMEDIAN POINT), X₃₇₁ (KENMOTU POINT), X₃₉ (BROCARD MIDPOINT), X₅₈₉ (2nd KENMOTU-VAN LAMOEN POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a/(a^{2} + 4 \operatorname{area}(ABC))\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(1.577634058995\)

▸ Linear Combinations1 combination

• Floor van Lamoen, Some concurrences from Tucker hexagons, :cite:`Forum Geometricorum <forum-geometricorum>` 2 (2002) 5-13.

▸ Incident Central Lines1 line

{39, 589}

▸ Functional Relations3 relations

• X₅₈₈ = isogonal conjugate of X₅₉₀

• X₅₈₈ = cevapoint of X₆ and X₃₇₁

• X₅₈₈ = X₁₉₉₄-cross conjugate of X₅₈₉

Notes

X₅₈₈ is the homothetic center of triangle ABC and the Lucas(2:1) homothetic triangle; see X₃₇₁ and X₅₈₉. (Randy Hutson, 9/23/2011)

Let A'B'C' be the Lucas central triangle. Let A″ be the trilinear pole of line B'C'; define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″; define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₅₈₈. (Randy Hutson, January 29, 2015)