Why three perpendiculars always sum to the same height

2026-05-13

Stand anywhere inside an equilateral triangle. Drop a perpendicular to each of the three sides. You now have three lengths - the distance from where you stand to each wall.

Walk somewhere else inside the triangle. The three lengths change. One grows, two shrink. One vanishes as you tiptoe up to a wall. But the sum of the three lengths doesn't move. It is exactly the triangle's height, and it does not care where you stand.

Drag the black dot. The blue, green, and red bars all change. The dashed outline next to them - the triangle's altitude - is the total they keep filling. Watching the bars rebalance while their column never overflows or underflows is the whole proof.

Why, in one line

Call the side length s and the three distances d₁, d₂, d₃. From the point, draw a line to each of the three corners. The triangle has just been split into three smaller triangles - one sitting on each side. Each of those small triangles has a side of length s (a side of the original) and a height equal to one of the ds.

area of the small triangle on side i = ½ s · d_i

The three small triangles fill the whole big triangle without overlap. So their three areas have to add up to the big triangle's area, which is ½ s h.

½ s(d₁ + d₂ + d₃)  =  ½ s h

Cancel the ½ s on each side. What's left is d₁ + d₂ + d₃ = h. The three distances must sum to the altitude, because the three little triangles must sum to the big one. The "must" is the proof.

What's it doing right at the edge?

Drag the dot until it touches one of the sides. The distance to that side is now zero. The other two distances absorb the slack, between them, exactly. Push the dot all the way into a corner and two of the distances vanish; the last one is the full altitude on its own.

The theorem doesn't care about boundary cases. The areas-sum argument keeps working: a side-triangle of zero height has zero area, and the rest pick up the rent.

Only equilateral, though

The cancellation worked because the three sides of the triangle all had the same length s. Try the same picture with a lopsided triangle and the three side-lengths are different, the ½ s won't factor out, and the sum of the perpendiculars genuinely does depend on where you stand.

Three equal sides is the secret. Equilateral triangles - and regular polygons more generally - have this kind of accountancy built in. Viviani noticed it in 1659 while studying with Torricelli. Three hundred and sixty seven years later, the argument still fits on a postcard.

Inside an equilateral triangle, the sum of the perpendicular distances from any point to the three sides equals the triangle's altitude. The proof is a single equation: three small triangles have to add up to the big one.