## X(1144) (EHRMANN CONGRUENT SQUARES POINT)

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           a/(a - L) : b/(b - L) : c/(c - L),
where L = L(a,b,c) is the smallest root of a2/(a - L) + b2/(b - L) + c2/(c - L) = 2D/L
where D = area(ABC).
Barycentrics    a2/(a - L) : b2/(b - L) : c2/(c - L)

Suppose P is a point inside triangle ABC. Let Sa be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define Sb and Sc cyclically. Then X(1144) is the unique choice of P for which the three squares are congruent. The function L(a,b,c) is symmetric, homogeneous of degree 1, and satisfies 0 < L(a,b,c) < min{a,b,c}. Also, X(1144) lies on the hyperbola {A,B,C,X(1),X(6)}; indeed, X(1144) lies on the open arc from X(1) to the vertex of ABC opposite the shortest side. L(a,b,c) is the common length of the sides of the three squares. (Jean-Pierre Ehrmann, 12/16/01)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.

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Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense