MHB Prove $BD=2CD$ in Triangle $ABC$

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In triangle $ABC$ with $AB=AC$, point $D$ lies on $BC$ and point $E$ on $AD$, where $\angle BED=2 \angle CED=\angle BAC$. The goal is to prove that $BD=2CD$. The discussion includes various approaches to the proof, emphasizing both trigonometric and geometric methods. Participants are encouraged to share their solutions and insights on the problem. The problem aims to engage the community in mathematical exploration and problem-solving.
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In a triangle $ABC$, it's given that $AB=AC$, point $D$ is on $BC$ whereas point $E$ is on $AD$ such that $\angle BED=2 \angle CED=\angle BAC$.

Prove that $BD=2CD$.

Note:

This problem is actually posted by Albert at another math forum about 2 years ago and for all information, I have gained his "permission" to post the exact same problem here, for the folks to have some fun solving it either trigonometrically or geometrically.
 

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My solution:
View attachment 2541

Since AB=AC, I first let $\angle CED=\alpha$, then $\angle EBC=\beta$ and $\angle DAC=\theta$.
A little working on the triangle $EBC$ shows that $\alpha+\beta+\theta=90^{\circ}$

Consider the triangles $EBD$ and $ECD$, we have:
$\dfrac{ED}{\sin \beta}=\dfrac{BD}{\sin 2\alpha}$ and $\dfrac{ED}{\sin(\beta-\alpha+2\theta)}=\dfrac{CD}{\sin \alpha}$

This yields
$\dfrac{BD}{CD}=\left(\dfrac{\sin 2\alpha}{\sin \alpha} \right)\left(\dfrac{\sin (\beta-\alpha+2\theta)}{\sin \beta} \right)$

We then try to eliminate the variable $\beta$ by using the relation $\beta=90^{\circ}-(\alpha+\theta)$, we get:

$\begin{align*}\dfrac{BD}{CD}&=\left(\dfrac{\sin 2\alpha}{\sin \alpha} \right) \left(\dfrac{\sin (90^{\circ}-(\theta+\alpha))-\alpha+2\theta)}{\sin (90^{\circ}-(\theta+\alpha))} \right)\\&=\dfrac{\sin 2\alpha}{\sin \alpha}\cdot\dfrac{\sin (90^{\circ}-(2\alpha-\theta))}{\sin(90^{\circ}-(\theta+\alpha))}\\&=\dfrac{\sin 2\alpha}{\sin \alpha}\cdot\dfrac{\cos (2\alpha-\theta)}{\cos (\theta+\alpha)}\\&=\dfrac{\sin 2\alpha}{\sin \alpha}\cdot\dfrac{\cos 2\alpha \cos\theta+\sin 2\alpha \sin \theta}{\cos \theta \cos\alpha-\sin \theta \sin \alpha}(*)\end{align*}$

We know the RHS expression will be reduced to 2, so, I think eliminating another variable would be a wise way to go.

Now, consider the triangles $ABE$ and $ACE$, we have:

$\dfrac{AB}{\sin(180^{\circ}- 2\alpha)}=\dfrac{AE}{\sin \theta}$ and $\dfrac{AC}{\sin(180^{\circ}- \alpha)}=\dfrac{AE}{\sin(\alpha-\theta)}$

Since $AB=AC$, the above simplifies to:

$\dfrac{\sin 2\alpha}{\sin \theta}=\dfrac{\sin \alpha}{\sin(\alpha-\theta)}$

$\dfrac{2\sin \alpha \cos \alpha}{\sin \alpha}\cdot\sin(\alpha-\theta)=\sin \theta$

$2\cos \alpha\cdot\sin(\alpha-\theta)=\sin \theta$

$2\cos \alpha\cdot(\sin \alpha \cos\theta-\cos \alpha \sin\theta)=\sin \theta$

$2\cos \alpha\cdot\sin \alpha \cos\theta=\sin \theta(1+2\cos ^2 \alpha)$

$\tan \theta=\dfrac{\sin 2\alpha}{1+2\cos ^2 \alpha}$ (**)

To merge these two equations ((**)and (*)), we divide the equation (*), top and bottom, by $\cos \theta$ to get:

$\dfrac{BD}{CD}=\dfrac{\sin 2\alpha}{\sin \alpha}\cdot \dfrac{\cos 2\alpha +\sin 2\alpha \tan\theta}{\cos \alpha -\sin \alpha \tan\theta}$

therefore,

$\begin{align*}\dfrac{BD}{CD}&=\dfrac{2\sin \alpha \cos \alpha}{\sin \alpha}\cdot \dfrac{\cos 2\alpha +\sin 2\alpha\cdot\frac{\sin 2\alpha}{1+2\cos ^2 \alpha}}{\cos \alpha-\sin \alpha\left(\dfrac{\sin 2\alpha}{1+2\cos ^2 \alpha} \right)}\\&=2 \cos \alpha\cdot \dfrac{\cos 2\alpha +2\cos^2 \alpha \cos 2\alpha+\sin^2 2\alpha}{\cos \alpha+2\cos^3 \alpha-\sin \alpha \sin 2\alpha}\\&=2 \cos \alpha\cdot \dfrac{\cos 2\alpha(1+2\cos^2 \alpha)+\sin^2 2\alpha}{\cos \alpha+2\cos^3 \alpha-2\sin \alpha \sin \alpha \cos \alpha}\\&=2 \cos \alpha\cdot \dfrac{\cos 2\alpha(\cos 2\alpha+2)+\sin^2 2\alpha}{\cos \alpha(1+2\cos^2\alpha-2\sin^2 \alpha)}\\&=2 \cos \alpha\cdot \dfrac{\cos^2 2\alpha+\sin^2 2\alpha+2\cos 2\alpha}{\cos \alpha(1+2(\cos^2\alpha-\sin^2 \alpha))}\\&=2 \cos \alpha\cdot \dfrac{1+2\cos 2\alpha}{\cos \alpha(1+2\cos 2\alpha)}\\&=2 \cos \alpha\cdot\dfrac{1}{\cos \alpha}\\&=2 \end{align*}$

Hence,

$BD=2CD$ (Q.E.D.)
 

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