Triangle Square Inscribing

There are two types of squares inscribing reference triangle DeltaABC in the sense that all vertices lie on the sidelines of ABC. 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 DeltaABC. Construct the perpendicular CD_|_AB and the line segment BE=AD. Bisect ∠BDC, and let F be the intersection of the bisector with BC. Then draw FK and FH through F, perpendicular to and parallel to AB, respectively. Let G be the intersection of FH and BC, and then construct FK and HJ through F and H perpendicular to AB. Then  square GHJI 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 BC, . Now join the new vertices S_(AB) and S_AC of this square with the vertex A, marking the points of intersection Q_(A,BC) and Q_(A,CB). Next, draw the perpendiculars to BC through Q_(A,BC) and Q_(A,CB). These lines intersect AB and AC respectively in Q_(AB) and Q_(AC). This results in the A^+-inscribed square Q_(A,BC)Q_(A,CB)Q_(AB)Q_(AC).

The triangle DeltaX^+Y^+Z^+ of centers of the A^+/--, B^+/--, and C^+/--inscribed squares form the inner inscribed squares triangle, which is perspective to DeltaABC with the outer Vecten point, Kimberling's X_(485), as its perspector.


A similar construction can be done by initially erecting a square internally on the side BC. This leads to the A^--inscribed square. The triangle DeltaX^-Y^-Z^- of centers of the A^--, B^--, and C^--inscribed squares form the outer inscribed squares triangle, which is perspective to DeltaABC with the inner Vecten point, Kimberling's X_(486), as its perspector.

The centers of the inscribed squares of type II are the intercepts of the orthic axis with the sides of DeltaABC. Consider the intercept X of the orthic axis and BC. The perpendicular to BC through X meets AB and AC in C_a, and B_a respectively. Together with points A^+ and A^- on BC these form the A-insquare of type II.

The lines connecting the vertices A_+B_+C_+ and A_-B_-C_- of these insquares are parallel to the orthic axis.

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

See also

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

Portions of this entry contributed by Floor van Lamoen

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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." Lamoen, F. "Friendship Among Triangle Centers." Forum Geom. 1, 1-6, 2001.Yiu, P. "Squares Erected on the Sides of a Triangle.", P. "On the Squares Erected Externally on the Sides of a Triangle."

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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.

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