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Solving a system of trig equations based on a diagram

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Based on a diagram (the numbers are just examples to aid in identification) I came up with the following equations:

$$ \begin{array}{l} ( x+y)\csc (\alpha )=( z+a)\csc (\alpha +\theta )\\ y\csc (\beta )=( z+a)\csc (\beta +\theta )\\ ( x+y)\csc (\gamma )=a\csc (\gamma +\theta )\\ y\csc (\delta )=a\csc (\delta +\theta ) \end{array} $$

which I reduced to

$$\theta = \arcsin\left(\frac{a}{\sec( \delta ) y}\right) -\delta =\arcsin\left(\frac{a}{\sec( \gamma )( x+y)}\right) -\gamma =\arcsin\left(\frac{z+a}{\sec( \beta ) y}\right) -\beta =\arcsin\left(\frac{z+a}{\sec( \alpha )( x+y)}\right) -\alpha$$

Solve for $x$ and eliminate $y$, $z$, and $\theta$. The remaining letters, $\alpha$, $\beta$, $\gamma$, $\delta$, and $a$, are constant. This is for a physical device, so all the lengths are positive, the angles except $\theta$ are between $0$ and $\frac\tau4=\frac\pi2=90^\circ$, $\theta$ is between $0$ and $\frac\tau2=\pi=180^\circ$, and close formula-based approximations are fine.

Mathematica (via the free plan for Wolfram Programming Lab) either timed out or ran out of memory: https://lab.wolframcloud.com/env/solly.ucko/SteepleMeasurer%201000.nb

SymPy said it couldn't solve this kind of equation:

Python 3.7.4 (tags/v3.7.4:e09359112e, Jul  8 2019, 19:29:22) [MSC v.1916 32 bit
(Intel)] on win32
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import *
>>> alpha, beta, gamma, delta, a, x, y, z, theta = symbols('alpha, beta, gamma,
delta, a, x, y, z, theta')
>>> parsed = parse_latex("\arcsin\left(\frac{a}{\sec( \delta ) y}\right) -\delta
 =\arcsin\left(\frac{a}{\sec( \gamma )( x+y)}\right) -\gamma =\arcsin\left(\frac
{z+a}{\sec( \beta ) y}\right) -\beta =\arcsin\left(\frac{z+a}{\sec( \alpha )( x+
y)}\right) -\alpha ")
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
NameError: name 'parse_latex' is not defined
>>> from sympy.parsing.latex import parse_latex
>>> parsed = parse_latex("\arcsin\left(\frac{a}{\sec( \delta ) y}\right) -\delta
 =\arcsin\left(\frac{a}{\sec( \gamma )( x+y)}\right) -\gamma =\arcsin\left(\frac
{z+a}{\sec( \beta ) y}\right) -\beta =\arcsin\left(\frac{z+a}{\sec( \alpha )( x+
y)}\right) -\alpha ")
Traceback (most recent call last):
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\Lexer.py", line 128, in nextToken
    ttype = self._interp.match(self._input, self._mode)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\atn\LexerATNSimulator.py", line 97, in match
    return self.matchATN(input)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\atn\LexerATNSimulator.py", line 126, in matchATN
    predict = self.execATN(input, next)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\atn\LexerATNSimulator.py", line 191, in execATN
    return self.failOrAccept(self.prevAccept, input, s.configs, t)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\atn\LexerATNSimulator.py", line 250, in failOrAccept
    raise LexerNoViableAltException(self.recog, input, self.startIndex, reach)
antlr4.error.Errors.LexerNoViableAltException: LexerNoViableAltException('')

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\sympy\parsing\latex\__init__.py", line 34, in parse_latex
    return _latex.parse_latex(s)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\sympy\parsing\latex\_parse_latex_antlr.py", line 85, in parse_latex
    relation = parser.math().relation()
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\sympy\parsing\latex\_antlr\latexparser.py", line 390, in math
    self.enterRule(localctx, 0, self.RULE_math)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\Parser.py", line 366, in enterRule
    self._ctx.start = self._input.LT(1)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\CommonTokenStream.py", line 61, in LT
    self.lazyInit()
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\BufferedTokenStream.py", line 186, in lazyInit
    self.setup()
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\BufferedTokenStream.py", line 189, in setup
    self.sync(0)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\BufferedTokenStream.py", line 111, in sync
    fetched = self.fetch(n)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\BufferedTokenStream.py", line 123, in fetch
    t = self.tokenSource.nextToken()
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\Lexer.py", line 130, in nextToken
    self.notifyListeners(e)             # report error
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\Lexer.py", line 285, in notifyListeners
    listener.syntaxError(self, None, self._tokenStartLine, self._tokenStartColum
n, msg, e)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\antlr4\error\ErrorListener.py", line 60, in syntaxError
    delegate.syntaxError(recognizer, offendingSymbol, line, column, msg, e)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\sympy\parsing\latex\_parse_latex_antlr.py", line 59, in syntaxError
    raise LaTeXParsingError(err)
sympy.parsing.latex.errors.LaTeXParsingError: I don't understand this
ight) -lpha csin\left(♀rac{z+a}{\sec( lpha )( x+y)}+y)}
^
>>> parsed = parse_latex("\arcsin\left(\frac{a}{\sec( \delta ) y}\right) -\delta
 =\arcsin\left(\frac{a}{\sec( \gamma )( x+y)}\right) -\gamma =\arcsin\left(\frac
{z+a}{\sec( \beta ) y}\right) -\beta =\arcsin\left(\frac{z+a}{\sec( \alpha )( x+
y)}\right) -\alpha ")
KeyboardInterrupt
>>> parsed = parse_latex("\\arcsin\\left(\\frac{a}{\\sec( \\delta ) y}\\right) -
\\delta =\\arcsin\\left(\\frac{a}{\\sec( \\gamma )( x+y)}\\right) -\\gamma =\\ar
csin\\left(\\frac{z+a}{\\sec( \\beta ) y}\\right) -\\beta =\\arcsin\\left(\\frac
{z+a}{\\sec( \\alpha )( x+y)}\\right) -\\alpha ")
>>> parsed
False
>>> ex1, ex2, ex3, ex4 = map(parse_latex, "\\arcsin\\left(\\frac{a}{\\sec( \\del
ta ) y}\\right) -\\delta =\\arcsin\\left(\\frac{a}{\\sec( \\gamma )( x+y)}\\righ
t) -\\gamma =\\arcsin\\left(\\frac{z+a}{\\sec( \\beta ) y}\\right) -\\beta =\\ar
csin\\left(\\frac{z+a}{\\sec( \\alpha )( x+y)}\\right) -\\alpha ".split("="))
>>> ex1, ex2, ex3, ex4
(-delta + asin(left(right*(a/((y*sec(delta)))))), -gamma + asin(left(right*(a/((
(x + y)*sec(gamma)))))), -beta + asin(left(right*((a + z)/((y*sec(beta)))))), -a
lpha + asin(left(right*((a + z)/(((x + y)*sec(alpha)))))))
>>> solve([ex1-ex2, ex2-ex3, ex3-ex4])
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\sympy\solvers\solvers.py", line 1173, in solve
    solution = _solve_system(f, symbols, **flags)
  File "C:\Users\Solly\AppData\Local\Programs\Python\Python37-32\lib\site-packag
es\sympy\solvers\solvers.py", line 1937, in _solve_system
    raise NotImplementedError('could not solve %s' % eq2)
NotImplementedError: could not solve -delta + gamma + asin(left(right*(a/((y*sec
(delta)))))) - asin(left(right*(a/(((x + y)*sec(gamma))))))
>>>
```
geometry trigonometry systems-of-equations
Original Q&A
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On BEST ANSWER

I'm going to replace $a$ with $w$, so that I can define $$a := \cot\alpha \qquad b := \cot\beta \qquad c := \cot\gamma \qquad d := \cot\delta \qquad t := \cot\theta \tag{0}$$

We can readily solve, say, the first three equations as a linear system in $x$, $y$, $z$ to get $$\begin{align} x &= w\,\frac{\sin(\alpha-\beta) \sin\gamma}{\sin(\gamma-\alpha)\sin(\beta + \theta)} &&= w\,\frac{b-a}{ (a-c) (b + t)\sin\theta} \\[4pt] y &= w\,\frac{\sin\beta \sin\gamma \sin(\alpha+\theta)}{\sin(\gamma-\alpha)\sin(\beta+\theta)\sin\theta} &&= w\,\frac{a + t}{(a-c)(b+t)\sin\theta} \\[4pt] z &= w\,\frac{\sin\alpha \sin(\gamma+\theta)}{\sin(\gamma-\alpha) \sin\theta} &&= w\,\frac{c+t}{a-c} \end{align}\tag{1}$$

If $w=0$, we're done. Otherwise, substituting these into the final equation, and dividing through by non-zero $w$ quickly yields

$$t = -\frac{a d - b c}{a-b-c+d} \tag{2}$$ so that $$b+t = -\frac{(a-b)(d-b)}{a-b-c+d} \qquad c+t = -\frac{(a-c)(d-c)}{a-b-c+d} \tag{3}$$ $$\sin\theta = \frac{1}{\sqrt{1+t^2}} = \frac{|a-b-c+d|}{\sigma} \qquad \sigma := \sqrt{(a-b-c+d)^2+(a d - b c)^2} \tag{4}$$ and then

$$\begin{align} x &= \sigma w\,\frac{\operatorname{sgn}(a-b-c+d)}{(c - a) (b - d)} \\[4pt] y &= \sigma w\,\frac{1}{(b - d)\,|a-b-c+d|} \\[4pt] z &= \phantom{\sigma}w\,\frac{c-d}{a-b-c+d} \end{align}\tag{$\star$}$$

It may be worth noting these relations:

$$\begin{align} (x:y) &= (a-b-c+d:c-a) \\[4pt] (w:z) &= (a-b-c+d :c-d) \end{align} \tag{5}$$

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