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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.

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

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...