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 .

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).