Given any △ABC (blue), you can construct three external equilateral triangles (green).
Napoleon's Theorem states that if you connect the centroids of the these triangles, the resulting △XYZ (red) will also be an equilateral triangle.
On the worksheet below, manipulate points A, B and C to see how changes △ABC affect the proportions of △XYZ. Use the guiding questions below to explore the properties of Napoleon's Theorem.
The following notes are essential for proving the theorem.

NL is rotated 30° clockwise around point B to yield N'L'.

ML is rotated 30° counterclockwise around point C to yield M'L''.

Both N'L' and M'L'' dilated by a k = 2*cos(30°) from points B and C respectively result in AP.

Because these three lines can be obtained through dilations, we know they are all parallel.

Further, because both are a rotation of 30° of the original segments, we can show that m∠NLM = 60°.

Given that △NLM is an isosceles and ∠NLM is NOT one of the base angles, we can prove that △NLM is equilateral.

Is the AREA of △NLM dependent on the AREA of △ABC? To explore this, see what happens when points A, B and C are co-linear.

The statement above says that the three green triangles should be EXTERNAL to △ABC. Does it appear to be necessary that they are external?

What happens to the ratios as you manipulate points A, B and C?