If A, B, C and D are four points in the plane such that | A B |^2+| C D |^2=| B C |^2+| D A |^2=100 , then A C B D =

Step-by-step solution

Let A, B, C and D are vertices of a quadrilateral So, | A C |^2+| B D |^2=| A B |^2+| C D |^2+2 B C A D and | A C |^2+| B D |^2=| A D |^2+| B C |^2+2 A B D C . From Eqs. (i) and (ii), aligned & B C A D = A B D C & ( O C - O B ) ( O D - O A )=( O B - O A ) ( O C - O D ) & OC OD - OC OA - OB OD + OB OA & = O B O C - O B O D - O A O C + O A O D & O C O D + O B O A = O B O C + O A O D & O C ( O D - O B )+ O A ( O B - O D )=0 & O C B D + O A D B =0 B D ( O C - O A )=0 & B D A C =0 aligned