536 Puzzles: Problem & Solution #284: Choosing a Site

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This is what the author called a trick question.  The sum of the distances are constant.  To see why draw in lines from the vertices of the original triangles forming subtriangles T1,T2,T3.  

Now the sum of the areas of T1,T2,T3 = area of original equilateral triangle which is constant for this problem.  Since the area of a triangle is Base * Height 

So it does not matter where you place the house.

I became curious about other types of triangles.  The solution for a 3-4-5 right triangle.