Question

$ABCD$ is a quadrilateral in which $AD=BC$. $E,F,G$ and $H$ are the mid-points of $AB,BD,CD$ and $AC$ respectively. Prove that $EFGH$ is rhombus.

Answer 1

Given that $AD=BC$ ......(1)
From the figure,
For triangle $ADC$ and triangle $ABD$
$2GH=AD$ and $2EF=AD$, therefore $2GH=22EF=AD$ ......(2)
For triangle $BCD$ and triangle $ABC$
$2GF=BC$ and $2EH=BC$, therefore $2GF=2EH=BC$ .......(3)
From $\left(1\right),\left(2\right),\left(3\right)$ we get,
$2GH=2EF=2GF=2EH$
$GH=EF=GF=EH$
Therefore $EFGH$ is a rhombus.
Hence proved