GeoProof — Viviani's Theorem explorer

d₁ + d₂ + d₃ = altitude h

The three colored lengths always fill exactly one altitude — drag P to see them trade off.

Viviani's Theorem

In an equilateral triangle, take any point P inside it and measure the perpendicular distance from P to each of the three sides. Those three distances always add up to the same total — the triangle's altitude (height) — wherever P happens to be:

d₁ + d₂ + d₃ = h

Slide P toward a side and that side's distance shrinks while the other two grow by exactly as much, keeping the sum fixed.

Why it's true

Join P to the three vertices, splitting the triangle into three smaller triangles, one sitting on each side. Each side has the same length s, so the three areas are ½·s·d₁, ½·s·d₂, ½·s·d₃. Together they fill the whole triangle, whose area is ½·s·h. Dividing by ½·s gives d₁ + d₂ + d₃ = h.

At special points

At the very center the three distances are equal, each one-third of the altitude. At a vertex two distances vanish and the third becomes the full altitude. On an edge, that edge's distance is zero and the other two still total h.

It needs an equilateral triangle

The clean version relies on all three sides being equal so the common factor ½·s cancels. For a general triangle the weighted sum of distances (each times its side length) is what stays constant instead.

A bit of history

It's named for Vincenzo Viviani (1622–1703), an Italian mathematician and a student of Galileo.