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The second part of this question about the scalene triangle remains a Tough Nut but Vassil, from Lawnswood High School, Leeds sent a good solution to the first part.



Vassils' diagram shows that the areas of the purple red and yellow triangles, for any position of $P$, add up to the area of the whole equilateral triangle. Taking the length of the side of the equilateral triangle to be $a$ units this gives: $${ah_1\over 2} + {ah_2\over 2} + {ah_3\over 2} = {ah\over 2}.$$ Hence $$h_1 + h_2 + h_3 = h = \rm{constant}.$$

The sum of the perpendicular distances from $P$ to the sides of the triangle is equal to the height of the triangle.