Draw ABC as an ISOSCELes triangle. On the side AB (lateral side) mark the mid point O. Now as O as the center, draw a circle with radius = OB =OA. It may not intersect base of all isosceles triangles. But we choose base BC of our circle LONG enough so that it will intersect BC (base) at D.
Now, OB = OA = OD = radius.
AB = 2 * radius = AC (isosceles triangle)
In triangle OBD, anle B = angle D as sides are equal. Since angle B = angle C, then angle B = angle C = angle D.
triangles OBD and ABC are similar. AB || OB, BD || BC. and angles are all equal.
as OB = 1/2 AB , BD = 1/2 BC.
Hence the proof is done.