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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.
1026411_8b80680288f94b53b2e69322f9543eac.png


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 proved

Mathematics

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