`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)` .

In A B C ,/_A=90^0a n dA D is an altitude. Complete the relation (B D)/(D A)=(A B)/(()) .

In A B C , on the side B C ,Da n dE are two points such that B D=D E=E Cdot Also /_A D E=/_A E D=alpha, then

In a triangle A B C ,Da n dE are points on B Ca n dA C , respectivley, such that B D=2D Ca n dA E=3E Cdot Let P be the point of intersection of A Da n dB Edot Find B P//P E using the vector method.

Let A B C D be a rectangle and P be any point in its plane. Show that A P^2+P C^2=P B^2+P D^2dot

In A B C , a point P is taken on A B such that A P//B P=1//3 and point Q is taken on B C such that C Q//B Q=3//1 . If R is the point of intersection of the lines A Qa n dC P , ising vedctor method, find the are of A B C if the area of B R C is 1 unit

ABC is a triangle and P any point on BC. if vec P Q is the sum of vec A P + vec P B + vec P C , show that ABPQ is a parallelogram and Q , therefore , is a fixed point.

A B C D is a tetrahedron and O is any point. If the lines joining O to the vrticfes meet the opposite faces at P ,Q ,Ra n dS , prove that (O P)/(A P)+(O Q)/(B Q)+(O R)/(C R)+(O S)/(D S)=1.

let P an interioer point of a triangle A B C and A P ,B P ,C P meets the sides B C ,C A ,A BinD ,E ,F , respectively, Show that (A P)/(P D)=(A F)/(F B)+(A E)/(E C)dot

D , E , F are three points on the sides B C ,C A ,A B , respectively, such that /_A D B=/_B E C=/_C F A=thetadot A^(prime), B ' C ' are the points of intersections of the lines A D ,B E ,C F inside the triangle. Show that are of A^(prime)B^(prime)C^(prime)=4cos^2theta, where is the area of A B Cdot

O P Q R is a square and M ,N are the middle points of the sides P Qa n dQ R , respectively. Then the ratio of the area of the square to that of triangle O M N is (a)4:1 (b) 2:1 (c) 8:3 (d) 7:3