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In DeltaABC, D is any point on side BC. ...

In `DeltaABC, D` is any point on side BC. Prove that
`AB+BC+CA gt 2AD`

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In the adjoining figure, D is any point on side BC. Join AD. We know that the sum of any two sides of a triangle is greater than the third side.
`:.` In `DeltaABD`
`AB+BD gt AD` ...(1)
and in `DeltaACD`
`AC+CD gt AD` ...(2)
Adding equations (1) and (2)
`AB+BD+AC+CD gt AD+AD`
`implies" "AB+(BD+CD)+AC gt 2AD`
`implies" "AB+BC+CA gt 2AD`
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