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`

In a triangle OAB,/_AOB=90^(0) .If P and Q are points of trisection of AB prove that OP^(2)+OQ^(2)=(5)/(9)AB^(2)

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 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 .

In A B C ,\ \ /_A=60o . Prove that B C^2=A B^2+A C^2-A B.A C .

If O is a point in space, A B C is a triangle and D , E , F are the mid-points of the sides B C ,C A and A B respectively of the triangle, prove that vec O A + vec O B+ vec O C= vec O D+ vec O E+ vec O Fdot

In a triangle ABC,N is a point on AC such that BN perp AC. If BN^(2)=ANNC, prove that /_B=90o .

In square A B C D ,\ P\ a n d\ Q are mid-point of A B\ a n d\ C D respectively. If A B=8c m\ a n d\ P Q\ a n d\ B D intersect at O , then find area of O P B

If A B is a chord of a circle, P\ a n d\ Q are the two points on the circle different from A\ a n d\ B , then /_A P B=\ /_A Q B /_A P B+/_A Q B=180^0\ or\ \ /_A P B=\ /_A Q B /_A P B+\ /_A Q B=90^0 /_A P B+\ /_A Q B=180^0

In Fig. 4.90, if P S Q R , prove that P O S ~ R O Q . (FIGURE)

A B C is a triangle, P is a point on A Ba n dQ is a point on A C such that /_A Q P=/_A B Cdot Complete the relation (A r e aof A P Q)/(A r e aof A B C)=dot(())/(A C^2) .