A Visual Proof of Viviani's Theorem

step

1

2

3

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.

Proof

Step 1. From an arbitrary point

P

, perpendiculars are drawn to each of the three sides. You can drag the locator at

P

.

Step 2. Rotate each of these segments around

P

by

±30°

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

P

.

Step 3. Translate the green triangle along

AC

by the vector

XC

, which translates

P

to

Y

. The three altitudes then project disjointly to the altitude of

ABC

.

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.

External Links

Viviani's Theorem (Wolfram MathWorld)

Viviani's Theorem

Permanent Citation

Tomas Garza, Jay Warendorff

"A Visual Proof of Viviani's Theorem"
http://demonstrations.wolfram.com/AVisualProofOfVivianisTheorem/
Wolfram Demonstrations Project
Published: October 16, 2020