Viviani's Theorem/Proof 2
Theorem
Let $T$ be an equilateral triangle.
Let $P$ be a point inside $T$.
Let $x$, $y$ and $z$ be the lengths of the perpendiculars dropped from $P$ to each of the three sides of $T$.
Then;
- $x + y + z = h$
where $h$ is the height of $T$.
Proof
Let $T = \triangle ABC$ be an equilateral triangle whose vertices are $A$, $B$ and $C$.
Let $h$ be the height of $T$.
Let $P$ be a point inside $T$.
Let $\triangle PDE$, $\triangle PFG$ and $\triangle PJH$ be three equilateral triangles constructed from $P$ to each side of $ABC$ as depicted.
Let the heights of $\triangle PDE$, $\triangle PFG$ and $\triangle PJH$ be $x$, $y$ and $z$.
Let $\triangle CGK$ be an equilateral triangle constructed also as depicted.
The result follows by inspection.
$\blacksquare$
Source of Name
This entry was named for Vincenzo Viviani.