Question

Plane Geometry
Prove that by connecting the midpoints of the sides of a convex quadrangle we get a parallelogram When is that parallelogram a rhombus ? A square ?

Answer 1

Here $ABCD$ is convex quadrilateral & $EFGH$ is polygen obtained by connecting mid-points of $ABCD$.

By joining $GE$ or $HF$, we get $2$ ${\Delta }_s$ and by mid-point theorem the segment containing $2$ mid-pts of adjacent sides is both parallel and half respective diagonals.

Now, as opp. two sides are parallel and equal so $EFGH$ is a $\parallel gm$.

When all the sides are equal, $EFGH$ is rhombus & when $EG$ & $HF$ bisect each other then square is formed.