Title
Inscribed Square
Problem Statement
Consider the four vertices of a square. To have a square inscribed in a triangle, two
of the square's vertices must lie on the same side of the triangle. Here are some
examples:
Given a triangle, can we construct an inscribed square? All inscribed squares?
Problem setup
Can a square be constructed inside a triangle in such a manner so that the vertices of the
square lie on the line segments of the triangle, resulting in two of the vertices lying on
the same line segment of the triangle? Can more than one square be constructed in the
triangle? Is there a relationship between the type of triangle and the number of squares
that can be constructed inside the triangle?
Plans to Solve/Investigate the Problem
My original plans were to construct a simple triangle using Geo-Sketchpad and try to
draw an inscribed square within.
Investigation/Exploration of the Problem
j = 1.20 cm
j' = 1.20 cm
j'' = 1.20 cm
j' D
j''
E
First, a simple acute triangle was constructed. Second, a point J” was identified on the
base of the triangle. Next, a perpendicular segment was constructed from the original
point to a point on the opposing side of the triangle. By marking the center point and
rotating that original segment two times, a perfect square was formed. Then it was
possible to move the vertices of the square until all 4 vertices were lying on the sides of
the triangle. Measurements of the sides of the square indicated that the square remained a
square. In order to determine how to predict where the point on the triangle would be as
a corresponding vertex for the square, a straight line was created through the vertex of the
triangle and the opposing vertex of the original square. That line passed through point J’
on the side of the triangle. Repeated similar constructions allowed for a prediction that
stated that the point on the triangle at which the vertex of the square would touch would
be through the vertex of the triangle to the other side of the triangle. Consequently, it
was possible to construct three inscribed squares inside the acute triangle. That number
does not hold true for all triangles, however, as the number of squares inscribed within an
obtuse triangle is limited to one.
j = 1.20 cm
j' = 1.20 cm
j'' = 1.20 cm
j' D
j''
E