MHB From Altitudes to Angles to Sides

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To find the sides of a triangle using Heron's formula, the area can be expressed in terms of the sides as A = 21a = 24b = 56c. By substituting the expressions for a, b, and c into Heron's formula, an equation in A can be derived. Solving this equation allows for the determination of the area A. Once A is known, the sides can be calculated from the altitudes. The proposed side lengths are 112/3*√3, 98/3*√3, and 14*√3.
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If you are allowed to use the Heron's formula, then you can do the following. Let $AB=c$, $BC=a$ and $AC=b$. Then the area of the triangle is $A=21a=24b=56c$. Express $a$, $b$ and $c$ though $A$ and substitute in the Heron's formula. You will get an equation in $A$, from where $A$ can be found. Then it is easy to find sides from altitudes.
 
Is it 112/3* \sqrt{3}, 98/3* \sqrt{3}and 14* \sqrt{3}
 
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