Jennifer Roth

Construct a circle of radius AB from a point on a line, and a a circle from another point C on the same line with radius AB (AC has to be greater than or equal to 2AB). Where these two circles intersect, is your third point for your isosceles triangle (there are two such points).

Click here to view the GSP file that demonstrates
this.

Now, how can we construct a parallelogram using this isosceles triangle.

If you reflect triangle ABC in AC you get the following parallelogram

Click here to view the GSP file that demonstrates
this. This parallelogram has sides of equal length.

It is a special kind of parallelogram, a rhombus, in which all four sides
are congruent and its diagonals are perpendicular.

Therefore, you would get a square if you started with a 45-45-90 triangle.

How could we produce a rectangle using a any right triangle. A simple construction
using perpendicular lines will produce the following if you start with triangle
ABC. (How else can we do this using reflections and translations?)

Now, how could we use any right triangle to produce a parrallelogram.

Click here to view the script that demonstrates
this. If you first reflect the triangle ABC in AC you get triangle ACD.
Then, if you reflect triangle BCD in DB you get the parallelogram BCDE.
This parallelogram, again, has equal sides. (How else can we do this?)

How can we construct a parallelogram using any triangle. Would this give
us a parallelogram that has different side lengths. Construct triangle ABC
and construct a line parallel to AC through B. Use this as your line of
reflection for triangle ABC. Then construct segment AA' and segment CC'.
You get a special parallelogram, another rectangle.

Click here to view the GSP file that demonstrates
this. How else can we use any triangle to produce a parallelogram?

Now, how do we construct any quadrilateral using triangles. Reflect along
one of the sides of the triangle, except for a right triangle, and you will
get a quadrilateral.

How do we construct a trapezoid using triangles.

Construct triangle ABC, and construct any point on AC. Then, construct
a line parallel to BC through this point, D. If you started with an isosceles
triangle, you would get an isosceles trapezoid. As the line segment ED approaches
length 0, the limit is your original triangle.

So, how can we construct other polygons using triangles. I will just focus
on regular figures (figures that are equilateral and equiangular).

As we know the measure of each interior angle of a regular n-gon is 1/n(n-2)(180
degrees). Therefore the pentagon above was constructed by rotating a line
segment 108 degrees, and the hexagon was constructed rotating a line segment
120 degrees, etc. As you can see the (n-2) is the number of triangles used
to make up each of the polygons.

Click here to view the GSP script used
to construct the Pentagon.

Click here to view the GSP script used to
construct the Hexagon.

Click here to view the GSP script used to
construct the Octagon.

What if we were to use an isosceles triangle with the base angle equal to
1/n(n-2)(180 degrees) and the other vertices at the center of the regular
n-gon? We could rotate this triangle the 1/n(n-2)(180 degrees) about the
center to produce the n-gon.