The elementary geometric prolbem is as below:
The length of $DE$ can be calculated to be exactly 7.
But the calculation is a little complicated with many steps via trigonometry and so on.
Is there any simpler method to prove that $DE=7$?
update
The method I am using, is mainly based on trigonometry, and polynomial system calculation.
For example,
first set up a Descartes coordinate system with origin at $D$, with $\vec{DA}$ as vertical and $\vec{DC}$ as horizontal axes;
then all coordinates of points $A, B, C$ can be easily obtained;
In order to obtain that of $E$, try to construct two circles as indicated in the figure above, with fixed circumferential angles on the specific chords; then one of the intersection points of the two circles is $E$.
Once coordinate of $E$ obtained, I can calculate the length of $DE$.
The whole process is just easy to understand but difficult to calculate. So far my calculation is based on symbolic software.
An algebraic approach: use the same Descartesian coordiante system as yours, then the coordinates:
$$A:(0,\dfrac{49}3),B:(-\dfrac{49}4,0),C:(\dfrac{49}4,0)$$
Let $E: (x,y)$, since $\cos\angle AED=-\dfrac{1}2$ and $\cos\angle BEC=-\dfrac{3}5$,
apply Cosine theorem to $\Delta BEC$ and $\Delta AED$ respectively so that you can obtain the following two equations:
$$\left\{ \begin{array}{rl} 6 x^2+6 y^2-98 y+\sqrt{\left(9 x^2+(49-3 y)^2\right) \left(x^2+y^2\right)}&=0 \\ 80 x^2+80 y^2+3 \sqrt{256 x^4+512 y^2 x^2-76832 x^2+256 y^4+76832 y^2+5764801}&= 12005 \\ \end{array} \right.$$
Visualize them and solve it (you can use any symbolic software, Maxima, Axiom, Mupad in matlab, maple, mathematica and so on):
there are two solutions to the equations:
$$x=\pm\dfrac{5 \sqrt{3}}{2},y=\frac{11}{2}$$
Which means $E$ can locate at either of the two positions inside $\Delta ABC$.
Then $DE=\sqrt{x^2+y^2}=7$