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}
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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