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Draw a triangle. On the edges of the triangle, construct equilateral triangles. Find the centroids of the equilateral triangles and connect them to form a new triangle.
- What is true about the new triangle?
- What happens when you reflect each centroid over the closest edge of your original triangle? What is the difference in areas between the outer and inner Napoleanic triangles?
- Now construct segments from the outside corners of the equilaterals triangles to the opposite vertices of the original triangle.
- What is true about these segments?
- What is true about these segments relative to the original triangle?
The following sketch is one "non-geometer's" first exploration of Napoleon's theorem.
Here's a link to this sketch by Sarah Seastone, for which you need The Geometer's Sketchpad.
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