Viviani's Theorem/Proof 2

Theorem

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 $T$.

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.

$\blacksquare$

This entry was named for Vincenzo Viviani.