Question

In triangle ABC,A=60 and ad is median,then prove 4ad*ad=c^2+b^2+bc

Answer 1

you extend the median AD by an equal amount DE = AD.    OR,

Draw CE at C parallel to AB. Draw BE parallel to AC. Then ABEC is a parallelogram.  The diagonal AE = 2 AD as the diagonals bisect each other in a parallelogram.  In the parallelogram, adjacent angles sum to 180deg. So angle ACC is = 180 - A  = 120 deg

So ACE form a triangle and the side AE is given by

AE² = AC² + CE² - 2 AC CE cos C
(2AD)² = b² + c² - 2 b c cos 120
4 AD² = b² + c² - 2 b c (-1/2)

hence the answer