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. In the related problem of finding a largest square contained in a triangle, not all four vertices are required to lie on the triangle's sides. Calabi's triangle is the unique triangle up to similarity, other than an equilateral triangle, with three distinct placements of a largest contained square. In two of these placements one vertex lies in the triangle's interior.
