I want to find angle x in this picture.

And this is what I’ve done so far. Without loss of generality, assume $overline{rm BC}=1$

then, $$overline{rm BD}= 2sin{frac{x}{2}}$$, $$overline{rm BH}= 4sin^2{frac{x}{2}}= 2(1-cos{x}), quad overline{rm CH} = 2cos{x}-1$$

$$overline{rm CE}=frac{2cos{x}-1}{sqrt{2-2cos{x}}}$$

Let $overline{rm DE}=y$,

since $bigtriangleup DCE = bigtriangleup HCE$,

$$frac{1}{2}ysin{50^{circ}}=frac{1}{2}sin{x}frac{(2cos{x}-1)^2}{2-2cos{x}}$$

Then by applying law of cosines to $bigtriangleup DEC$,

$$y^2+1-2ycos{50^{circ}}=frac{(2cos{x}-1)^2}{2-2cos{x}}$$

So we have a system of equations

$$begin{cases}ysin{50^{circ}}=sin{x}frac{(2cos{x}-1)^2}{2-2cos{x}}\y^2+1-2ycos{50^{circ}}=frac{(2cos{x}-1)^2}{2-2cos{x}} end{cases}$$

But it’s too messy to solve since 50 is not special angle.

How can I solve this problem?