B4. A flag in the shape of an equilateral triangle is hung from two vertices at the tops of two vertical poles. One pole has a length of 4 and the other pole a length of 3. It is also given that the third vertex of the flag touches the ground exactly.

Determine the length of the side of the flag.

![](https://cdn.mathpix.com/cropped/2024_04_17_3d028e43617be6eb4b2ag-2.jpg?height=508&width=583&top_left_y=1959&top_left_x=1162)
B4. The length of the equilateral triangle $x \cdot x^{2}=|B D|^{2}+1^{2}=(|E A|+|A F|)^{2}+1=$

$\left(\sqrt{x^{2}-4^{2}}+\sqrt{x^{2}-3^{2}}\right)^{2}+1=2 x^{2}-25+2 \sqrt{x^{2}-16} \times \sqrt{x^{2}-9}+1$.

$2 \sqrt{x^{2}-16} \times \sqrt{x^{2}-9}=-x^{2}+24$. Squaring: $4\left(x^{4}-25 x^{2}+144\right)=x^{4}-48 x^{2}+576$ or $3 x^{4}-52 x^{2}=0$.

Since $x \neq 0$, we find $3 x^{2}=52$. Thus, $x=\sqrt{\frac{52}{3}}=2 \sqrt{\frac{13}{3}}=\frac{2}{3} \sqrt{39}$.
![](https://cdn.mathpix.com/cropped/2024_04_17_608b2a812c4856940d57g-2.jpg?height=734&width=1488&top_left_y=1780&top_left_x=314)

(C) Stichting Nederlandse Wiskunde Olympiade 2007