Three Distances Problem

Inspired by post by Glenn:

http://mathforum.org/epigone/geometry-puzzles/chansmyrpir /3suu2jldc1ax@legacy

Subject: area of a triangle given the distance of a point from




 the 3 vertices inside the triangle
Author: glenn gilongos <morpheus_ibangil@yahoo.com>
Date: 11 Mar 04 05:59:10 -0500 (EST)
<skip>
A point is inside an equilateral triangle. 
If its distances from the 3
vertices are 3, 4 ,5 units, 
what is the area of the equilateral triangle?




 *********
It can be shown that, if a point may be inside as well as outside equilateral triangle, then there are two triangles in question,




with sides equal to 
 
 and  
 
In the Figure, not the best disposition of two (the larger blue and the lesser red) triangles is shown. 




Sure each triangle can be rotated around the green point, 

and even flipped. 
It seems that in almost all cases there are two solutions.