congruence of two triangles

Question

if two corresponding sides of two triangles are equal and their medians drawn on the third sides are also equal, prove two triangles are congruent. I tried to solve it by extending the median by its size and make a parallelogram but it didn't help.

Answer 1

Let in $BD$ be median of $\Delta ABC$ and $QS$ be a median of $\Delta PQR$,

where $AB=PQ$, $BC=QR$ and $BD=QS$.

We'll prove that $\Delta ABC\cong\Delta PQR$.

Indeed, let $ABCE$ and $PQRT$ be parallelograms.

Thus, $AE=BC=QR=PT$ and $BE=2BD=2QS=QT$, which says $\Delta ABE\cong \Delta PQT$,

which gives $\measuredangle ABD=\measuredangle PQS$, $\Delta ABD\cong\Delta PQS$, $AD=PS$ and we are done!