If D is the midpoint of AC, then AD = DC....and AC = 2(AD)

In right triangle ABC,

(BC)^2 = (AB)^2 +(2(AD))^2

(BC)^2 = (AB)^2 +4(AD)^2 -----------(i)

In right triangle ABD,

(BD)^2 = (AB)^2 +(AD)^2 ----------(ii)

Eq,(i) minus Eq.(ii),

(BC)^2 -(BD)^2 = 3(AD)^2

Oops...it is not proven.