Sum of lengths of perpendicular to the sides of an equilateral triangle

#EquilateralTriangle#Perpendiculars

Problem Statement

Prove that the sum of lengths of perpendicular from an interior point $P$ to the sides of an equilateral triangle is equal to the altitude of the equilateral triangle.

Let $ABC$ be an equilateral triangle such that $P$ is an interior point. Let $p_{1}, p_{2}$ and $p_{3}$ be the lengths of perpendicular from $P$ upon $BC, AB$ and $CA$ respectively. Let the side length of the equilateral triangle be $a$. Since, we have:

$$ \text{area}(\Delta PBC ) + \text{area}(\Delta PBA )+ \text{area}(\Delta PAC ) = \text{area}(\Delta ABC ) $$

$$ \implies \frac{1}{2}(p_{1}+p_{2}+p_{3})a = \frac{\sqrt{3}}{4}a^2 $$

$$ \implies p_{1}+p_{2}+p_{3} = \frac{\sqrt{3}}{2} a = \text{length of the altitude of the equilateral triangle.} $$