X₁₁₄₄

EHRMANN CONGRUENT SQUARES POINT

Suppose that P is a point inside triangle ABC. Let \(S_{A}\) be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define \(S_{B}\) and \(S_{C}\) cyclically. Then X₁₁₄₄ 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, X₁₁₄₄ lies on the hyperbola {A, B, C,X₁,X₆}; indeed, X₁₁₄₄ lies on the open arc from X₁ 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)

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

X₁₁₄₄ (EHRMANN CONGRUENT SQUARES POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(a/(a - L) : b/(b - L) : c/(c - L)\), where L = L(a,b,c) is the smallest root of \(a^{2}\)/(a - L) + \(b^{2}\)/(b - L) + \(c^{2}\)/(c - L) = 2D/L, and \(D = \operatorname{area}(ABC)\)

Barycentrics: \(a^{2}/(a - L) : b^{2}/(b - L) : c^{2}/(c - L)\)

Search Value (6, 9, 13): \(1.872006645063\)

Notes

See Jean-Pierre Ehrmann, Congruent Inscribed Rectangles.