Question

if ad is perpendicular to bc and bd is 1/3 cd prove that 2(ca)^2=2(ab)^+(bc)^2

Answer 1

ABD is a right angled triangle 
and AB is the hypotaneous

(AB)² = (AD)² + (BD)²

(AB)² - (BD)² = (AD)² ---------(i)


now,
ACD is also a right angled triangle 
(AC)² = (AD)² + (DC)²
(AC)² - (DC)² = (AD)² ----------(ii)

(i) = (ii) 

(AB)² - (BD)² =(AC)² - (DC)²
(AB)² +(DC)² = (AC)² + (BD)²