`B D` and `C E` are bisectors of `/_B` and `/_C` of an isosceles ` A B C` with `A B=A Cdot` Prove that `B D=C Edot`
Since `AB = AC angleABC = angleACB ……....(i)` [Angles opposite to equal sides are equal]

Explanation

Since `AB = AC angleABC = angleACB ……....(i)` [Angles opposite to equal sides are equal]
Since `BD `and `CE` are bisectors of `angle B` and `angle C `
`angle ABD = angle DBC = angle BCE = ECA = angleB/2 = angleC/2 …......(ii)`
Now, Consider triangleEBC = triangle DCB
`angleEBC = angle DCB`......... [From (i)]
`BC = BC `......[Common side]
`angle BCE = angleCBD.........` [From (ii)]
By `ASA` congruence criterion, `triangle EBC ≅ triangleDCB`
Since corresponding parts of congruent triangles are equal.
`CE = BD `
or, `BD = CE`
Hence proved.