Question

if in triangle abc ad is a median and AE perpendicular to BC that prove that a b square + AC square equal to 2AD square + half BC square

Answer 1

in triangle ABC, AD is median

then , BD = BC

in triangles ABD and ADC

AB2 = AD2 + BD2 .........(1)

AC2 = AD2 + BC2 ............(2)

now add both the equations,

AB2 + AC2 = AD2 + AD2 + BD2+ BC2

AB2 + AC2 = 2AD2 + 2BD2 [ since BD = BC]

AB2 + AC2 = 2[AD2 + BD2]

hence proved...

HOPE THIS WOULD HELP YOU OUT

Answer 2

AB^2= AE^2+BE^2
AC^2=AE^2+EC^2

Adding both equations
AB^2+AC^2= AE^2+BE^2+AE^2+EC^2
AB^2+AC^2= 2AE^2+BE^2+EC^2
AB^2+AC^2= 2(AD^2-DE^2) +BE^2+EC^2
AB^2+AC^2= 2AD^2-2DE^2+BE^2+EC^2
(BE=BC)
AB^2+AC^2= 2AD^2-2DE^2+2BE^2
AB^2+AC^2= 2AD^2+2(BE^2-DE^2)
AB^2+AC^2= 2AD^2+2(BD^2)
AB^2+AC^2= 2AD^2+2(1/2BC^2)^2
AB^2+AC^2= 2AD^2+1/2BC^2
Therèfore proved