Viviani's Theorem/Proof 1

Proof
Let $T = \triangle ABC$ be an equilateral triangle whose vertices are $A$, $B$ and $C$.

Let $h$ be the height of $T$.

Let $a$ be the length of one side of $T$.

Let $P$ be a point inside $T$.

Let $\triangle APB$, $\triangle BPC$ and $\triangle CPA$ be the three triangles formed by joining $P$ to each of the three vertices $A$, $B$ and $C$ of $T$.

Let the heights of $\triangle APB$, $\triangle BPC$ and $\triangle CPA$ be $x$, $y$ and $z$.


 * Vivianis-Theorem.png

By definition, these heights are the lengths of the perpendiculars dropped from $P$ to each of the three sides of $T$.

Let $A$ be the area of $T$.

By Area of Triangle in Terms of Side and Altitude:
 * $A = \dfrac {a h} 2$

But we also have that the area of $T$ is also equal to the sum of the areas of each of $\triangle APB$, $\triangle BPC$ and $\triangle CPA$.

By Area of Triangle in Terms of Side and Altitude, these areas are equal to $\dfrac {a x} 2$, $\dfrac {a y} 2$ and $\dfrac {a z} 2$.

That is:


 * $A = \dfrac {a h} 2 = \dfrac {a x} 2 + \dfrac {a y} 2 + \dfrac {a z} 2$

from which it follows that:
 * $h = x + y + z$