Given a point $P$ outside equilateral $\Delta ABC$ but inside $\angle ABC$, if the distance between $P$ to $BC,CA,AB$ are $h_1,h_2,h_3$ respectively, where $h_1 – h_2 + h_3 = 6$, find $[\Delta ABC]$ .
What I Tried: At first I couldn't understand if $h_1,h_2,h_3$ are just any lines touching the sides or are some specific lines like altitudes or medians (bisecting the sides of the triangle) . But since they are denoted like $h_1,h_2,h_3$ I suppose they are the altitudes. So here is a picture :-
No idea for this problem. I don't think I can use any simple geometry techniques here like angle-chasing, area of triangles, pythagorean theorem and so on, because I have been given very less info. So I am a bit stuck here.
Can anyone help me? Thank You.
Best Answer
By Viviani's theorem, the height of $\triangle ABC$ is $6$, so its side length is $\frac{12}{\sqrt3}=4\sqrt3$ and its area is $\frac{6×4\sqrt3}2=12\sqrt3$.