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.

Solution

## 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 $$(1), (2), (3)$$ we get,$$2GH=2EF=2GF=2EH$$$$GH=EF=GF=EH$$Therefore $$EFGH$$ is a rhombus.Hence provedMathematics

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