prove that two line segments are congruent (have the same length)

Question

Problem. In the diagram above, prove that $DE=DN$.

The reason I'm asking is because this is the main ingredient to answering this question.

It seems like it should be a simple high school geometry problem, but I can't seem to crack it. Any help would be much appreciated.

Thanks!

Answer 1

Notice that when $GN=GL=x$ tends to a really small value, $DN$ and $DE$ will become part of a right angle triangle $\Delta DNE/\Delta DLE$. As the length of the hypotenuse and a side cannot be the same, the condition is false.