Viviani's Theorem ... EQUILATERAL triangles

The Italian mathematician Vincenzo Viviani (April 5, 1622 – September 22, 1703) gave his name to a theorem applicable to EQUILATERAL triangles - i.e. triangles with all three sides the same length.

Viviani's Theorem

states that the sum of the shortest distances from any interior point (P) to the sides of an equilateral triangle equals the length of the triangle's altitude. 

In an equilateral triangle, all three sides are of equal length, and consequently, all three altitudes are also equal in length. Each altitude in an equilateral triangle not only serves as a perpendicular segment from a vertex to the opposite side but also acts as a median and an angle bisector.

By applying the half-base x height formula for the Area of a Triangle, the theorem is easily proved.

Proof 

△ ABC is equilateral with side length a units and altitude of length h units. P is a randomly chosen point inside the triangle.

Lines are drawn from P to each of the vertices A, B, and C. Three triangles are formed PAB, PBC, and PCA. Let the vertical height of those triangles be s, t and u respectively.

The areas of those triangles are ${\displaystyle \frac{u\cdot a}{2}}$, ${\displaystyle \frac{s\cdot a}{2}}$, and ${\displaystyle \frac{t\cdot a}{2}}$. 

They exactly fill the enclosing triangle, so the sum of these areas is equal to the area of the enclosing triangle. So we can write:

${\displaystyle \frac{u\cdot a}{2}+\frac{s\cdot a}{2}+\frac{t\cdot a}{2}=\frac{h\cdot a}{2}}$

and thus

${\displaystyle u+s+t=h}$

The converse is also true - If the sum of the distances from an interior point of a triangle to the sides is independent of the location of the point, the triangle is equilateral.

Links

Wikipedia - Viviani's Theorem

Dynamic Mathematics Learning - 2D generalisations of Viviani's Theorem

Gogeometry.com - Viviani's Theorem - Equilateral Triangle 

David Darling - Viviani's theorem  - "Viviani's theorem is that for a given point inside an equilateral triangle, the sum of the perpendicular distances from the point to the sides is equal to the height of the triangle. If the point is outside the equilateral triangle, the relationship still holds if one or more of the perpendiculars is treated as a negative value."

James Tanton - Viviani's Theorem - Video - This also looks at the cases where point P is outside the equilateral triangle.

Mind Your Decisions - Viviani's Theorem 

Cut the Knot - Viviani.