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If h, j, and k represent the lengths of the altitudes of a triangle, the area of the triangle is given by the equation:
A=(hjk)^2/sqrt(2(hjk)^2(h^2+j^2+k^2)-((hj)^4+(hk)^4+(jk)^4))
The above formula is derived by solving simultaneously the six pythagorean equations involving the altitudes and the segments of the sides they partition.
Letting h=4, j=5, and k=6 --> A=3600/sqrt(57239) --> the sides have lengths 1200/sqrt(57239), 1440/sqrt(57239), and 1800/sqrt(57239).
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