Thales Theorem with Trapezoids

Question

I'm solving a math problem, I figured out most of the coordinates, but I can't find the 2 perpendicular sides (marked in green).

I know I need to use Thales Theorem.

Can anyone help, please?

Answer 1

I don't know how Thales is supposed to be useful. But here is how to solve it by trigonometry. With apologies to the French tourism industry, I've stripped the irrelevant information from the picture, and labeled the points:

You may wonder about my removing the Grotte Célestine, but if the questioned length is only to it rather than all the way to the Croix de Bacchus, the problem is clearly unsolvable, as there is no information given that would locate Grotte Célestine on the $\overline {FE}$ line. So I am assuming that the length desired is $FE$.

Drop a perpendicular from $C$ to $\overline{FE}$, intersecting it at $G$.

Now $\tan \angle ACB = \frac 54$, so $m\angle ACB \approx 0.896$ radians, and similarly $m\angle DCE \approx 0.211$. because the other three angles of $\square BCGF$ are right, so must be $\angle BCG$. Which means $$m\angle ECG \approx \frac \pi 2 - 0.211 - 0.896 \approx 0.464$$

So $$\cos \angle ECG \approx 0.894\\\sin \angle ECG \approx 0.447$$ and $$BF = CG = 14.3 \cos \angle ECG \approx 12.8$$ And $$GE = 14.3 \sin \angle ECG \approx 6.4\\FE = FG + GE = 12.8 + 6.4 = 19.2$$