The Midpoint Theorem of Triangle

$ABC$ is a triangle right-angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and Parallel to $BC$ intersects $AC$ at $D$. Show that $CM=MA=\frac{1}{2}AB$

$ABC$ is a triangle right-angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and Parallel to $BC$ intersects $AC$ at $D$. Show that $MD\perp AC$.

$ABC$ is a triangle right-angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and Parallel to $BC$ intersects $AC$ at $D$. Show that $D$ is the midpoint of $AC$

Show that the line segments joining the midpoints of the opposite sides of a quadrilateral and bisect each other.

In a parallelogram $ABCD,E,$and $F$ are the midpoints of the sides $AB$ and $DC$ respectively. Show that the line segment $AF$ and $EC$ trisect the diagonal $BD$.

Show that the figure formed by joining the midpoints of sides of a rhombus successively is a rectangle.

$ABCD$ is quadrilateral $E, F, G$ and $H$ are the midpoints of $AB, BC, CD$ and $DA$ respectively. Prove that $EFGH$ is a parallelogram.

$ABC$ is a triangle. $D$ is a point on $AB$ such that $AD=\frac{1}{4}AB$ and $E$ is the point on $AC$ such that $AE=\frac{1}{4}AC$. If $DE=2 cm$ find $BC$.