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# Triangle Square Inscribing

There are two types of squares inscribing reference triangle in the sense that all vertices lie on the sidelines of . The first type has two adjacent vertices of the square on one side, the second type has two opposite vertices on one side.

Casey (1888, pp. 10-11) illustrates how to inscribe a square in an arbitrary triangle . Construct the perpendicular and the line segment . Bisect , and let be the intersection of the bisector with . Then draw and through , perpendicular to and parallel to , respectively. Let be the intersection of and , and then construct and through and perpendicular to . Then is an inscribed square. Permuting the order in which the vertices are taken gives an additional two congruent squares. These squares, however, are not necessarily the largest inscribed squares. Calabi's triangle is the only triangle (besides the equilateral triangle) for which the largest inscribed square can be inscribed in three different ways.

An alternative construction is to externally erect a square on the side , . Now join the new vertices and of this square with the vertex , marking the points of intersection and . Next, draw the perpendiculars to through and . These lines intersect and respectively in and . This results in the -inscribed square .

The triangle of centers of the -, -, and -inscribed squares form the inner inscribed squares triangle, which is perspective to with the outer Vecten point, Kimberling's , as its perspector.

A similar construction can be done by initially erecting a square internally on the side . This leads to the -inscribed square. The triangle of centers of the -, -, and -inscribed squares form the outer inscribed squares triangle, which is perspective to with the inner Vecten point, Kimberling's , as its perspector.

The centers of the inscribed squares of type II are the intercepts of the orthic axis with the sides of . Consider the intercept of the orthic axis and . The perpendicular to through meets and in , and respectively. Together with points and on these form the -insquare of type II.

The lines connecting the vertices and of these insquares are parallel to the orthic axis.

The circle through , and is the -Apollonius circle (of type 3).

Ehrmann Congruent Squares Point, Lucas Circles, Square, Square Inscribing, Triangle

Portions of this entry contributed by Floor van Lamoen

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## References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Coxeter, H. S. M. and Greitzer, S. L. "Points and Lines Connected with a Triangle." Ch. 1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1-26 and 96-97, 1967.van Lamoen, F. "Inscribed Squares." Forum Geom. 4, 207-214, 2004.van Lamoen, F. "Vierkanten in een driehoek: 1. Omgeschreven vierkanten." http://home.wxs.nl/~lamoen/wiskunde/vierkant.html.van Lamoen, F. "Friendship Among Triangle Centers." Forum Geom. 1, 1-6, 2001.Yiu, P. "Squares Erected on the Sides of a Triangle." http://www.math.fau.edu/yiu/bottema38.pdf.Yiu, P. "On the Squares Erected Externally on the Sides of a Triangle." http://www.math.fau.edu/yiu/square.pdf.

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Triangle Square Inscribing

## Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Triangle Square Inscribing." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangleSquareInscribing.html