Thread: Square

There is an $\displaystyle ABCD$ square with side length of 1. $\displaystyle E\in\overline{AB}$ and $\displaystyle F\in\overline{BC}$ so that the perimeter of triangle $\displaystyle EBF$ is 2. Show that $\displaystyle EDF\angle =45^\circ$.

Hi

I have found a solution but don't know if it is the simplest one !

Let x=BE and y=BF
We know that $\displaystyle x+y+\sqrt{x^2+y^2} = 2$ (perimeter of triangle EBF)

On one hand
$\displaystyle \vec{DE}.\vec{DF} = \sqrt{1+(1-x)^2} \times \sqrt{1+(1-y)^2} \times \cos(\hat{EDF})$

On the other hand
$\displaystyle \vec{DE}.\vec{DF} = (\vec{DA}+\vec{AE}).(\vec{DC}+\vec{CF}) = (1-y)+(1-x) = 2-x-y$

Combining the 2 latter relations and using $\displaystyle x+y+\sqrt{x^2+y^2} = 2$ you can show that $\displaystyle \cos^2(\hat{EDF}) = \frac{1}{2}$