Answer :

Given:- ΔABC, AD is median

To Prove:- If AD is perpendicular on base BC then ΔABC is isosceles



Proof:- Let, A at Origin


be position vector of B and C respectively


Therefore,



Now position vector of D, mid-point of BC i.e divides BC in 1:1


Section formula of internal division: Theorem given below


Let A and B be two points with position vectors


respectively, and c be a point dividing AB internally in the ration m:n. Then the position vector of c is given by


Position vector of D is given by



Now distance/length of BC


= position vector of C-position vector of B



Now, assume median AD is perpendicular at BC


Then by Dot Product







AC = AB


Thus two sides of ΔABC are equal


Hence ΔABC is isosceles triangle


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