A point $P$ is selected inside an equilateral triangle. If sum of lengths of perpendicular dropped on to sides from $P$

Question

Problem :

A point $P$ is selected inside an equilateral triangle. If sum of lengths of perpendicular dropped on to sides from $P$ is $2014$, then $\frac{\mathrm{length\; of \; altitude \; of \; triangle}}{2014}$ is equal to

I have no clue how to proceed in this .. please help.. greatful to you.. thanks..

Answer 1

Check out the proof of Viviani's theorem. It doesn't matter where you select $P$ in an equilateral triangle, the sum of the perpendiculars is a constant. By Viviani's (or imagining that you've selected one of the vertices as your point $P$, 2014 is exactly the altitude of the triangle, so the ratio is 1.