X589

2nd KENMOTU-VAN LAMOEN POINT

X589 is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of X372 and the points where the squares in the Kenmotu configuration with center X372 meet sideline BC, and B' and C' are defined cyclically. See the reference at X588.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 39, 588 }: Central Line L(39, 588) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7148, 0.4568) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_588 = 1st KENMOTU-VAN LAMOEN POINT (0.5810, 0.2407) X_589 = 2nd KENMOTU-VAN LAMOEN POINT (0.6340, 0.4099) A B C X6 X372 X39 X588 X589 (a, b, c) = (5, 6, 7) | AB = 1

X589 (2nd KENMOTU-VAN LAMOEN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

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): \(1.133404162873\)
▸ Incident Central Lines1 line

{39, 588}

▸ Functional Relations3 relations

Notes

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

Let A'B'C' be the Lucas(-1) 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 X589. (Randy Hutson, January 29, 2015)