Napoleon's Triangle Calculator
Construct equilateral triangles on each side
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Napoleon's Theorem
Construct equilateral triangles on each side. Their centroids form an equilateral triangle!
Napoleon's Triangle
Understanding Napoleon's Theorem
Napoleon's Theorem is a beautiful result in geometry: If equilateral triangles are constructed on the sides of any triangle, the centroids of those equilateral triangles form another equilateral triangle!
The Theorem
Given any triangle ABC:
- Construct equilateral triangles on each side (either all outward or all inward)
- Find the centroids of these three equilateral triangles
- These centroids form an equilateral triangle called the Napoleon Triangle
Two Napoleon Triangles
Outer Napoleon
Equilateral triangles constructed outward from the original triangle.
Inner Napoleon
Equilateral triangles constructed inward toward the original triangle.
Area Relationship
A remarkable result: The difference between the outer and inner Napoleon triangle areas equals the area of the original triangle:
Area(Outer) - Area(Inner) = Area(Original)
Frequently Asked Questions
Did Napoleon actually discover this?
While it's named after Napoleon Bonaparte (who was indeed interested in mathematics), the theorem was likely known before him. The attribution is uncertain but the name stuck.
Why is the Napoleon triangle equilateral?
This follows from the symmetric construction - each centroid is positioned consistently relative to the original triangle's sides, creating equal spacing.
What if the original triangle is equilateral?
If the original triangle is equilateral, the Napoleon triangles have the same center as the original, and the inner Napoleon reduces to a point!
How do you find the centroid of an equilateral triangle?
The centroid is the average of the three vertices: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). It's also where the medians intersect.