O is any point inside a rectangle ABCD. Prove that `O B^2+O D^2=O A^2+O C^2`.

O is any point inside a triangle A B C . The bisector of /_A O B ,/_B O C and /_C O A meet the sides A B ,B C and C A in point D ,E and F respectively. Show that A D * B E * C F=D B * E C * F A

A point O in the interior of a rectangle A B C D is joined with each of the vertices A ,\ B ,\ C and D . Prove that O B^2+O D^2=O C^2+O A^2

O is any point in the interior of A B Cdot Prove that A B+A C > O B+O C A B+B C+C A > O A+O B+O C O A+O B+O C >1/2(A B+B C+C A)

In a triangle O A B ,\ /_A O B=90^0dot If P\ a n d\ Q are points of trisection of A B prove that O P^2+O Q^2=5/9A B^2dot

Let O be a point inside a triangle A B C such that /_O A B=/_O B C=/_O C A=omega , then show that: cotomega=cotA+cotB+cot C Cos e c^2omega=cos e c^2A+cos e c^2B+cos e c^2C

In a triangle O A B ,/_A O B=90^0dot If P and Q are points of trisection of A B , prove that O P^2+O Q^2=5/9A B^2

A B C D is a parallelogram and O is any point in its interior. Prove that: (i) a r\ ( A O B)+a r\ (C O D)=1/2\ a r(A B C D) (ii) a r\ ( A O B)+\ a r\ ( C O D)=\ a r\ ( B O C)+\ a r( A O 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 E^2+C D^2+B F^2

O is any point on the diagonal B D of the parallelogram A B C D . Prove that a r( /_\O A B)=a r( /_\O B C)

O is any point on the diagonal B D of the parallelogram A B C D . Prove that a r( triangle O A B)=a r(triangle O B C)