Equilateral triangles

Question

Let $ABC$ be a triangle with $AB = 1$, $AC = 2$ and $m(\widehat{BAC}) = 30^\circ$. We build on the outside the equilateral triangles $ABM$ and $ACN$.

Let $D$, $E$ and $F$ be the midpoints of $AM$, $AN$ and $BC$.

a) Prove that triangle $DEF$ is equilateral.

b) Find $x$ and $y$ so that $\overrightarrow{MN} = x\overrightarrow{AB} + y\overrightarrow{AC}.$

For a) all I found is that $DE = \frac{\sqrt{5 + 2\sqrt{3}}}{2}$. For b) I have no idea how I should start.

Thanks in advance!

Answer 1

This is very easy using coordinates. Notice that $AB\perp AN$, so we can set $A=(0,0)$, $B=(0,1)$, $N=(2,0)$, $M=(-\sqrt3/2,1/2)$, $C=(1,\sqrt3)$.

It is then straightforward to find the coordinates of midpoints $D$, $E$, $F$ and check that $DE=EF=FD$, as well as finding $x$ and $y$ such that $\overrightarrow{MN} = x\overrightarrow{AB} + y\overrightarrow{AC}$.

It would be nice to prove that $DEF$ is equilateral by a purely geometrical argument. I haven't succeeded yet, but I noticed that this result has a more general validity: it is true for any value of $\angle BAC$, and for any length of $AB$ and $AC$.

Answer 2

Let G be the midpoint of AC.

By midpoint theorem and some angle chasing, we have $\triangle DAE \cong \triangle FGE$.

Then, ED = EF and $\angle DEF = 60^0$ (from the fact that $\alpha = \alpha’$ and using $\angle DEG$ as the middle man).