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Viviani's theorem states that in an equilateral triangle, the sum of the distances from any interior or boundary point to the three sides is equal to the altitude of the triangle.

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Proof

Step 1. From an arbitrary point , perpendiculars are drawn to each of the three sides. You can drag the locator at .

Step 2. Rotate each of these segments around by and extend the segments to the corresponding side. This produces three equilateral triangles colored red, blue and green. The vertical line in each triangle is an altitude, equal to the altitude from .

Step 3. Translate the green triangle along by the vector , which translates to . The three altitudes then project disjointly to the altitude of .

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Contributed by: Tomas Garza (December 2020)
Based on an idea by: Jay Warendorff
Open content licensed under CC BY-NC-SA


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References

[1] C. Alsina and R. B. Nelsen, Charming Proofs: A Journey into Elegant Mathematics, Washington, D.C.: Mathematical Association of America, 2010 p. 96.

[2] A. Bogomolny. "Viviani's Theorem." Cut the Knot. (Oct 6, 2020) www.cut-the-knot.org/Curriculum/Geometry/VivianiPWW.shtml#explanation.



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