Prove that the quadrilateral formed by joining the midpoints of the sides of a rectangle is a rhombus, but not a square.

The medians $BE$ and $CF$ of the $∆ABC$ intersect each other at $G$ and the line segment $EF$ intersects $AG$ at $O$. Prove $OA=3OG$

$ABCD$ is a trapezium in which $AB\parallel DC$ and $E$ is the mid-point of $AD$. If $F$ be a point on $BC$ such that $EF\parallel AB$, then prove that $EF=\frac{1}{2}(AB+DC)$.

The line segment $EF$ obtained by joining the mid-points $E$ and $F$ of the sides $AB$ and $AC$ respectively of the $∆ABC$, is extended to $D$ such that $EF=FD$. Prove that $ADCE$ is a parallelogram.

If $\mathrm{E}, \mathrm{F}$ and $\mathrm{G}$ are the midpoints of the sides $\mathrm{AB}, \mathrm{BC}$ and $\mathrm{CA}$ of the $\mathrm{ABC}$ respectively, then prove that the centres of gravity of $\mathrm{ABC}$ and $\mathrm{EFG}$ are the same point.

Prove that if two medians of triangle be equal, then it is an isosceles triangle.

 In the trapezium $ABCD$, $AB\parallel DC$. The diagonals $AC$ and $BD$ of it intersects each other at $O$. Prove that $∆AOD=∆BOC$.

$D, E$ and $F$ are the midpoints of the sides $AB$. $AC$ and $BC$ of the $∆ABC$. Prove that $DF$ and $AE$ bisects each other.

 $E$ is the midpoint of the median $AD$ of the $∆ABC$. Extended $BE$ intersects $AC$ at $F$. Prove that $AF=\frac{1}{3}AC$.