# Problem on Equilateral Triangle and points

Equilateral $\triangle{ABC}$ with sides $2\sqrt{3}$. Let $P$ be the point outside$\triangle{ABC}$ such that points $A$ and $P$ lie opposite to $BC$. Let $PD$, $PE$, $PF$ be the perpendicular dropped on side $BC$, $AC$ and $AB$ receptively where $D$, foot of perpendicular, lies inside the segment $BC$. Let $PD=2$. How to find $PE+PF$.

The sum of perpendiculars is altitude $H$ in case of an internal point P. If the point P lies outside and has length of nearest perpendicular $h$ then the total of perpendicular distances is:

$$altitude \,H + h = 3 +2 =5,$$

which can be verified by drawing perpendiculars on all three sides and calculating extra length parts in $PE,PF$,over and above that for point $P$ when $on$ nearest side $BC.$ Shown in red color in a rough sketch $PD=h$ below.

EDIT1:

Angles $$DPE = DPF = 60^0$$

Red length total = $$h (\cos 60^0 + \cos 60^0) = 2 ( \frac12 + \frac12) =2$$

• how to calculate red part $=2$ ? Jul 17 '16 at 3:46
• Please see above edit. Jul 17 '16 at 4:57