__Essay #3: Boxes Within Boxes and Triangles
Within Triangles__

First construct a square (script: **square**).

Then connect each vertex to the midpoints of
the opposite sides (script: ** midpoint-square**).
As shown in the following figure, the construction lines generate
a square. It is this square that we wish to find the volume of
the smaller red square in the middle in relation to the given
square.

The ratio of the small, red square to the large
red square can be found by algebra or geometry. However, we will
use __technology__ to determine the ratio as a visual proof
in order to assist the student in __visualizing__ the problem.

First, if we allow the computer to calculate
the area of the small, interior box (GHIJ) over the area of the
larger box (ABA'B'), the ratio is .20 (and its reciprocal is 5,
meaning the __larger__ box is 5 times the area of the smaller
box).

Secondly, by using parallel construction lines,

In a more general case, we can choose Point
I along segment AB. With computerization (using Geometric Sketch
Pad, GSP, program), we can calculate the inside square, the outside
square, and their ratios depending upon the distance that Point
I is located along segment AB from Point A (ie, a **variable**
distance).

To further illustrate the relationship, an
animation is available where the point follows a path along segment
**AB**. (To activate this animation, **CLICK
HERE**.)

As the animation will reveal two characteristics of this construction:

1. The

closerPoint I is to Point A, thelargerArea KLMN is. In fact, it approaches Area AB'A'B.2. The

fartherPoint I is from Point A, thesmallerArea KLMN is. In fact, it approaches zero.

As a further exploration in technology, let us look at the triangle ABC and construct an internal triangle using medians (lines from vertices to the midpoint of the opposite side), similar to what we did in the above square.

Obviously, when the medians are drawn, there
is **no internal triangle**, since the medians meet at a common
point called the centroid (See Points G, H, and I). (A similarly
constructed triangle is available for personal exploration by
using the following **script**.)

Now, if we move Point D along the line BC at
variable distances from Point B (while the other points E and
F ** are moving proportionally** along their respective
line segments), then we will obtain a triangle GHI within the
original triangle ABC.

As a parenthetical remark, triangle GHI is
**not similar** to triangle ABC since angle BAC is __not__
congruent to angle GIH.

Now we can either **manually** move Point
D along the segment BC or **animate** along the same path while
while Points E and F are moved in a similar manner along their
respective triangle sides. Then we can observe how the triangle
GHI is formed and changes shape depending upon how far Point D
moves along Segment BC. (Click ** CLICK
HERE** for your own

If you took the time to explore with the above
animation opportunity, you would get similar results as shown
above and below. Notice in the **above figure** that when **Point
D** is to the ** left of the midpoint** of line BC,
triangle GHI is formed with

If **Point D** is to the ** right
of the midpoint**, then the internal triangle flips over so
that

As a parenthetical remark, there will be a
maximum, internal triangle when Point D is co-located either at
Point B or Point C. This last remark should be intuitive. If it
is not intuitive, then it is recommended to conduct the personal
exploration as offered above. (** CLICK
HERE** to go to the above place