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.
Do not write an equation yet. Shake the figure first.
Pull the point toward every side
Each distance grows or shrinks, while their sum stays locked to the triangle’s height.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
Any point P inside an equilateral triangle; perpendicular distances d₁, d₂, d₃ to the sides.
Return to the exact question
Show that from any interior point of an equilateral triangle, the three perpendicular distances to the sides sum to the triangle’s height.
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.
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