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Ehrmann Congruent Squares Point

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EhrmannCongruentSquaresPoint

Consider a point P inside a reference triangle DeltaABC, construct line segments AP, BP, and CP. The Ehrmann congruent squares point is the unique point P such that three equal squares can be inscribed internally on the sides of DeltaABC such that they touch the line segments in exactly two points each.

The side lengths of these triangles are given by the smallest root of the cubic equation

(a^2)/(a-L)+(b^2)/(b-L)+(c^2)/(c-L)=(2Delta)/L,
(1)

and the center function is

alpha_(1144)=a/(a-L),
(2)

which is Kimberling center X_(1144).

L(a,b,c) is symmetric, homogeneous of degree 1, and satisfies

L(a,b,c)<min(a,b,c).
(3)

X_(1144) lies on the (nonrectangular) circumhyperbola ABCX_1X_6.

See also

Kenmotu Point, Triangle Square Inscribing

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References

Ehrmann, J.-P. "Congruent Inscribed Rectangles." Forum Geom. 2, 15-19, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200203index.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(1144)=Ehrmann Congruent Squares Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1144.

Referenced on Wolfram|Alpha

Ehrmann Congruent Squares Point

Cite this as:

Weisstein, Eric W. "Ehrmann Congruent Squares Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EhrmannCongruentSquaresPoint.html

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