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Home / Geometry / If D is the middle point of the side BC of ∆ABC, prove that AB+AC>2AD.

If D is the middle point of the side BC of ∆ABC, prove that AB+AC>2AD.

General enunciation: If D is the middle point of the side BC of ∆ABC, prove that AB+AC>2AD.

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Particular enunciation: Suppose, in triangle ∆ABC, D is the mid-point of BC. Let us join (A, D). Let us prove that AB+AC>2AD.

Construction: Let us expand AD up to DE such that, AD = DE. (E, C) are joined.

Proof: In triangle ∆ABD and ∆CDE we get,

BD = CD                             [∵ D is the mid-point of BC]

AD = DE                             [∵ D is the mid-point of BC]

and ∠ADB =∠CDE                 [vertically opposite angle]

So the two triangles are congruent

∴ AB = CE

Now in ∆ACE,   AC+CE>AE

⟹AC+AB>AD+DE   [∵AB =CE]

⟹AB+AC>AD+AD   [∵ DE = AD]

⟹AB+AC>2AD           (proved)

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