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 let the circle you draw side on AB of triangle ABC intersects the base BC at a point E 
and the center of the circle you draw be O . then 
so triangle OBE is isosceles and 
angle OBE = angle ABC 

there fore triangles ABC and OBE are similar.
but (OB/AB)=1/2 

therefore (BE/BC)=1/2 
there fore E divides BC into two 

hence proved....

The diagram is given in the pic below

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