Proving a triangle is isoceles

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In the graphic we have an isosceles triangle, and the problem is

Calculate $\text{m}\angle BCD$

I added the point $E$ at distance $x$ from $C$ because it causes $DE=x$, after playing with geogebra. With this, the question is easily solved. Of course since the triangle is determined (Since the ratios are scale invariant, we can WLOG assume $x=1$)we can prove $DE=x$ using trigonometry, but what is the elegant, geometric-like way of showing it?

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1 Answer

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I post a solution for a similar question that I had solved in the past:

In this problem $x=10$. In your case the answer will be $80-10=70$. You draw $AEC$ equilateral triangle and see similarity between $ABE$ and $CDA$. In you case you would draw an equilateral triangle on $AC$ side but first join $C$ and $D$.

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