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.
