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In Triangle ABC, medians BE and CD are produced respectively to points X and Y such that CD=DX and BE=EY as shown in the figure. Show that the points X, A and Y are collinear. Also show that A is the mid point of XY

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Consider triangles AEY and CEB.
AE=CE (BE bisects AC)
EY=EB (median is extended its own length)
angles AEY and CEB are congruent (vertical angles)

So, the two triangles are congruent (SAS). That means that AY=BC and is parallel.

Now do the same with triangles ADX and BDC.

The rest follows.
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