Given a triangle , an inscribed square is a square all four of whose vertices lie on the edges of and two of whose vertices fall on the same edge. As noted by van Lamoen (2004), there are two types of squares inscribing reference triangle in the sense that all vertices lie on the sidelines of . In particular, the first type has two adjacent vertices of the square on one side, while the second type has two opposite vertices on one side. There are three squares of each type, and van Lamoen (2004) gives centers and vertices of the three squares of each type in homogeneous barycentric coordinates.

An inscribed square of type I can be obtained by externally erecting a square on the one of the sides, say . 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 (van Lamoen 2004).

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).

Consider the lengths of all possible squares inscribed in a non-obtuse triangle as discussed above. Inspection suggests that the side lengths of all such squares are very close to each other, an observation confirmed by Oxman and Stupel (2013), who showed that if are the lengths of the sides of any two such squares, then .

Casey (1888, pp. 10-11) gives a geometric construction for one type of square inscription on an arbitrary triangle as follows. 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.

Note that these squares are not necessarily the *largest possible* 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.