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Nick's Mathematical Puzzles: Solution 117
Simple enough, right?
Sum of the 3 heights = 10sqrt(3) / 2 = ~8.66
So your missing height = ~8.66 - 1 - 3 = ~4.66
the area of the equilateral triangle is immediately known. call the vertices of the equilateral triangle as A.B,C and join P to A,B and C. the triangle is subdivided into three sub-triangles out of which the area of two of the sub-triangles is known since the length of the altitudes are known. the third unknown one can be known by equating the area of the equilateral to the sum of the areas of the three sub-triangles.