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Let A B C be an isosceles triangle with ...

Let `A B C` be an isosceles triangle with `A B=A C` and let `D , E ,F` be the mid-points of `B C ,C A` and `A B` respectively. Show that `A D_|_F E` and `A D` is bisected by `F Edot`

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Let AD intersect FE at M. Join DE and DF.
Now, D and E being the midpoints of the sides BC and CA respectively, we have
DE||AB and `DE=(1)/(2) AB " " ` (by midpoint theorem).
Similarly, DF||AC and `DF = (1)/(2) AC. `
`therefore AB = AC rArr (1)/(2) AB =(1)/(2) AC rArr DE=DF. " " `...(i)
Now, DE||FA and `DE = FA " " [ because "DE||AB and "DE=(1)/(2) AB =FA]`
`rArr " DEAF is a ||gm " rArr "DEAF is a rhombus " [ because DE=DF " from (i), " DE = FA and DF = EA].`
But, the diagonals of a rhombus bisect each other at right angles.
`therefore AD bot FE and AM= MD.`
Hence, ` AD bot FE ` and AD is bisected by FE.
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