In a triangle `A B C ,\ A C > A B` , `D` is the mid-point of `B C` and `A E_|_B C` . Prove that: (i) `A B^2=A D^2-B C D E+1/4B C^2` (ii) `A B^2+A C^2=2\ A D^2+1/2B C^2`

In figure, AD is a median of a triangle ABC and A M_|_B C . Prove that: (i) A C^2=A D^2+B C.D M+ ((B C)/2)^2 (ii) A B^2=A D^2-B CdotD M+ ((B C)/2)^2 (iii) A C^2+A B^2=2A D^2+ 1/2B C^2

In figure, ABC is a triangle in which /_A B C=90^@ and A D_|_B C . Prove that A C^2=A B^2+B C^2-2B C.B D .

In A B C , A D is a median. Prove that A B^2+A C^2=2\ A D^2+2\ D C^2 .

In Figure, ABD is a triangle right-angled at A and A C_|_B D . Show that (i) A B^2=B C.B D (ii) A C^2=B C.D C (iii) A D^2=B D.C D

In figure, ABC is triangle in which /_A B C > 90o and A D_|_C B produced. Prove that A C^2=A B^2+B C^2+2B C.B D .

In figure, O is a point in the interior of a triangle ABC, O D_|_B C ,O E_|_A C and O F_|_A B . Show that (i) O A^2+O B^2+O C^2-O D^2-O E^2-O F^2=A F^2+B D^2+C E^2 (ii) A F^2+B D^2+C E^2=A W^2+C D^2+B F^2

A B D is a right triangle right-angled at A and A C_|_B D . Show that A B^2=B CdotB D (ii) A C^2=B CdotD C (iii) A D^2=B DdotC D (iv) (A B^2)/(A C^2)=(B D)/(D C)

A B C D is a parallelogram and E is the mid-point of B C ,\ D E\ a n d\ A B when produced meet at Fdot Then A F= 3/2A B (b) 2\ A B (c) 3\ A B (c) 5/4\ A B

A B C is a triangle. D is a point on A B such that D=1/4A B\ a n d\ E is a point on A C such that A E=1/4A Cdot Prove that D E=1/4B C

In a quadrilateral A B C D , given that /_A+/_D=90o . Prove that A C^2+B D^2=A D^2+B C^2 .