# triangle related challenge

In triangle $ABC$, $BA=BC$ and $\angle B=90^\circ$. $D,E$ are the points on $AB,BC$ respectively such that $AD=CE$. $M,N$ are points on $AC$ such that $DM$ is perpendicular to $AE$ and $BN$ is perpendicular to $AE$. Prove that $MN=NC$.

How can we solve this geometric problem by application of midpoint theorem?

Midpoint theorem states that in triangle ABC if D,E are midpoints of AB,AC respectively then AB is parallel to BC and AB =BC/2.

And converse of midpoint theorem states that:- In triangle ABC if D is the midpoint of AB and a line say U is drawn passing through D parallel to BC then U intersects AC(Say at E) and A-E-C and AE=EC.

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I added a sketch that should make your question a bit easier to grasp. However, I'm still puzzled about How can we solve this sum by application of midpoint theorem? What sum? And maybe it's my lack of acquaintance with English geometry terminology, but what is the midpoint theorem? –  t.b. May 23 '12 at 10:36
SUm(meaning:- this particular question.) –  meg_1997 May 23 '12 at 13:43
Midpoint theorem states that in triangle ABC if D,E are midpoints of AB,AC respectively then AB is parallel to BC and AB =BC/2. –  meg_1997 May 23 '12 at 13:45
and converse of midpoint theorem states that:- In triangle ABC if D is the midpoint of AB and a line say U is drawn passing through D parallel to BC then U intersects AC(Say at E) and A-E-C and AE=EC –  meg_1997 May 23 '12 at 13:49
So I wanted to say that whether we can prove the required result of this geometric problem using the above two theorems –  meg_1997 May 23 '12 at 13:51

You need to continue the line containing $BC$ beyond $B$ till the intersection with $DM$. Call the point of intersection $P$ (I chose to connect $A$ and $P$, though this is not strictly necessary).
Now $\angle PDB = \angle AEB$ as the angles whose sides are mutually perpendicular. $\angle PBD = \angle ABE = \pi/2$. $\Delta ABC$ is isosceles, hence $AB=BC$. $AD=EC$ by assumption, therefore $BD=BE$. Thus $\Delta AEB= \Delta PDB$.
Therefore, $PB=AB=BC$. $PM\parallel BN$ since they are both $\perp$ $AE$. Now you can invoke the converse midpoint theorem for $\Delta PMC$ and line $NB$.
Third paragraph should read $\triangle ABC$ is isosceles, not equilateral. –  Zander Jun 6 '12 at 2:04