Equilateral Triangle Perpendiculars

For an equilateral triangle of side $a$, the sum of perpendicular distances $p_1 + p_2 + p_3$ from any interior point to the sides is equal to the altitude $h$ of the triangle. Thus, $h = s$. Since height of an equilateral triangle is $h = \frac{\sqrt{3}}{2} a$, we have $a = \frac{2s}{\sqrt{3}}$. The area of the equilateral triangle is: $\text{Area} = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} \left(\frac{2s}{\sqrt{3}}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{4s^2}{3} = \frac{s^2}{\sqrt{3}}$