Viviani’s Theorem — Three Distances Always Sum to the Height
Pick any point inside an equilateral triangle: the three distances to the sides always add to the same value — exactly the triangle’s height. Move the point and nothing changes.
Problem
Show that from any interior point of an equilateral triangle, the three perpendicular distances to the sides sum to the triangle’s height.
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Any point P inside an equilateral triangle; perpendicular distances d₁, d₂, d₃ to the sides.
Why this approach works
Connect P to the three vertices: the triangle splits into three small triangles, each with base s (a side) and height equal to P’s distance to that side. Their areas sum to the whole: ½s·d₁ + ½s·d₂ + ½s·d₃ = ½s·h. Cancel ½s and get d₁ + d₂ + d₃ = h — independent of where P sits.
Proof
The three triangles from P share base s with heights d₁, d₂, d₃; their areas sum to the whole ½s·h, giving d₁+d₂+d₃ = h, independent of P.
Try it yourself
Slide P toward a vertex: one distance nears 0 while the others grow, yet the three still sum to the height h.
Related mathematics
Beyond MathVoyage
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