Napolean’s Triangle</center>
<center>Investigation# Essay #2

**Napolean's Triangle**

by: Ashley Little and Michelle Lowry

**The Construction**

Given triangle ABC, Napolean's triangle is constructed in the following
way:

Construct equilateral triangles on each side of the triangle ABC.

Find the centers of each constructed equilateral triangle by finding the angle
bisectors of each angle.

Connect the midpoints of each of the equilateral triangles to form Napoleon's
triangle.

**The Investigation**

It appears that this newly constructed triangle is equilateral.
Animation using GSP confirms this conjecture. If the original triangle is
equilateral, then Napoleon's triangle is also equilateral.

Consequently, the larger triangle formed by the constructed triangles is also
equiateral. If Napoleon's triangle is dialated by a scale factor of 2, the
larger triangle is formed.

Next, we tried to extend the concept beyond Napoleon's triangle to polygons
with more than three sides. The first attempt was with a general
quadrilateral. When a general quadrilateral is sketched, and squares are
constructed on each side of the quadrilateral there was no apparrent
relationship between the figure formed by the centers of the constructed
squares.

Next, we investigated special quadrilaterals. When the quadrilateral is a
parallelogram, the figure constucted by connecting the centers of the squares
constructed on each side of the parallelogram is a square.