Rectangle with two Equilateral Triangles

Question

Let $ABCD$ be a rectangle and let $ABE, BCF$ be equilateral triangles built, one outside, and one inside the rectangle. How do I compute $|EF|$ given the side lengths of the rectangle.

Any Hints?

Thanks

Answer 1

Hint:

$\triangle EBF$ is right at $B$.

Answer 2

Hint 2: Your drawing is misleading, for the vertex of the inner triangle will not always lie directly above the vertex of the outer triangle. But if we call the height $h$ and the width $w$, then the vertical distance from $E$ to the horizontal line on which $F$ lies is $\frac{\sqrt{3}}{2}w + \frac{h}{2}$. Can you figure out the horizontal distance from $F$ to the vertical line on which $E$ lies? It's something rather similar. Then Pythagoras does the rest.

Answer 3

Consider $60^{\circ}$ clockwise rotation around the point $B$.

Is it clear that then point $E$ is mapped to $A$ and $F$ is mapped to $C$? Then what can you say about triangles $EBF$ and $ABC$?

Answer 4

Triangle $BFE$ is right-angled triangle.