Question

In triangle abc d is a median then show that ab2+ac2=2(ad2+bd2)

Answer 1

From right triangle ABE

(AB)²=(AE)²+(BE)² ..................................(1)

From right triangle ACE

(AC)²=(AE)²+(EC)² .................................(2)

From right triangle ADE

(AD)²=(AE)²+(ED)² .................................(3)

Adding equation (1) & (2)

(AB)²+(AC)²=(AE)²+(BE)²+(AE)²+(EC)²

(AB)²+(AC)²=2(AE)²+(BE)²+(EC)²

(AB)²+(AC)²=2(AE)²+(BD-ED)²+(ED+DC)²

(AB)²+(AC)²=2(AE)²+(BD)²+(ED)²-2(DB)(ED)+(ED)²+(DC)²+2(ED)(DC)

(::DC=BD)

(AB)²+(AC)²=2(AE)²+2(ED)²+(BD)²+(BD)²-2(BD)(ED)+2(ED)(BD)

From equation (3)

(AB)²+(AC)²=2(AD)²+2(BD)²

(AB)²+(AC)²=2(AD²+BD²)

HENCE PROOVED.

HOPE THIS WILL HELP YOU.