Question
$AB$ is chord of circle $O$,points $D$ and $E$ are chosen on $AB$ in a way that $AD=BE$.prove two chords that are perpendicular to $AB$ and pass $D$ and $E$ points are equal.(prove $LK=MN$)
we are not allowed to use Similarity,ceva,power of point in circle.
things i have done so far:
$MN$ and $KL$ are parallel because $EN$ is perpendicular to $AB$ and $DK$ is perpendicular to $AB$.
acr $LM$ and arc $KN$ are equal because $MN$ and $KL$ are parallel.
my idea is to prove $BEN$ and $ADK$ are congruent then prove $BNN$ and $AKL$ are congruent.
for proving that $BEN$ and $ADK$ are congruent i need $BN = AK$ or one of those acute angles equality.
and i stuck.
UPDATE
well using user7000 idea, if i prove $LN$ and $KM$ are diameter then the problem is solved.
so we draw $LN$ and $KM$.they intersect each other at $O'$. we can say $\angle LNM =\angle KMN$.so $O'M = O'N $.with same way we can conclude $O'L = O'K$.if show $KO'=O'M$ then the problem is solved.
Converting comments to answer, as requested.
The perpendicular from $O$ to $\overline{KL}$ must also be perpendicular to $\overline{MN}$. (Why?) Let $P$ and $Q$ be the points where the perpendicular crosses these segments.
Show that $\overline{OP}\cong\overline{OQ}$. (Hint: The perpendicular from $O$ to $\overline{AB}$ meets the segment at its midpoint, which is also the midpoint of $\overline{DE}$. (Why?))
Then $\triangle OPK$, $\triangle OPL$, $\triangle OQM$, and $\triangle OQN$ are all congruent (via Hyptenuse-Leg), and the result follows.