Suppose you pick a point in an equilateral triangle. What would the sum of the distances from that point to the sides of the triangle, where the distances are defined as length drawn from the perpendicular of 1 side length, be? The answer is the height of the triangle!
Definitions: Let’s define a side of a triangle as y Let’s define the heights of triangles APC, BPC, APB as h1, h2, h3 respectively.
Using the formula of the area of a triangle, which is 1/2 of base x height:
Thus, ½(h1 x y) + ½(h2 x y) + ½(h3 x y) = Area( whole triangle) which is the sum of the little triangles.
Therefore, ½(h1 x y) + ½(h2 x y) + ½(h3 x y) = ½(y x total height of original triangle)
Factorising out ½, we obtain (h1 x y) + (h2 x y) + (h3 x y) = y x total height of original triangle
Factorising out y, we obtain h1 + h2 + h3 = total height of original triangle!
numberdot 