Question 872841
In triangles CBD & ABD

angle CBD is congruent to angle ABD
side BC is congruent to side AB ( isosceles triangle)
AD is the common side

so the triangles are congruent

there fore AD = DC

therefore D is the mid point of AC

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OR you can also prove by angular bisector theorem

BD is the angular bisector of angle C
AC=BC

D is any point on the angular bisector of angle C

Any point on the angular bisector of and angle is equidistant from the endpoints of the sides of the angle contained by them

therefore D is the mid point of AB