In an isoceles triangle ABC with AB=AC, D and Eare points on BC such that BE=CD.Show that AD=AE.

Consider the triangle ΔABC, points D and E can be put on side BC in two ways.

Let's say we put E close to B and D close to C,

then the triangles ABE and triangle ACD are SAS congruent,

since AB = AC........given,

B = C......angles opposite to equal sides,

and BE = CD ... given,

Hence AD = AE.

Now let's put E close to C and D close to B,

then in triangle ABD and ACE are SAS congruent,

since AB = AC........given,

B = C......angles opposite to equal sides,

And BD = CE, as shown below,

BE = CD

BE - DE = CD - DE

BD = CE,

Hence AD = AE.

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