`A B C` is a triangle, `P` is a point on `A B a 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)/(()) .

A B C is a triangle with /_B greater than /_C ,Da n dE are points on B C such that A D is perpendicular to B Ca n dA E is the bisector of angle Adot Complete the relation /_D A E=1/2()+/_C]dot

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.

In triangle 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 , using vector method, find the area of A B C if the area of B R C is 1 unit

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 triangle A B C ,ifPdotQ ,R divides sides B C ,A C , and A B , respectively, in the ratio k :1(inor d e r)dot If the ratio ((a r E A P Q R)/(a r e a A B C)) IS 1/3,t h e nk is equal to 1/3 (b) 2 (c) 3 (d) none of these

ABC is a right-angled triangle of which /_A = right angle. P and Q are two points on AB and AC respectively.By joining P,Q,B,Q and C,P prove that BQ^(2) + PC^(2) = BC^(2) + PQ^(2)

Let A B C be a triangle with A B=A Cdot If D is the midpoint of B C ,E is the foot of the perpendicular drawn from D to A C ,a n dF is the midpoint of D E , then prove that A F is perpendicular to B Edot

D , E ,a n dF are the middle points of the sides of the triangle A B C , then (a) centroid of the triangle D E F is the same as that of A B C (b) orthocenter of the tirangle D E F is the circumcentre of A B C

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 B in D ,E ,F , respectively, Show that (A P)/(P D)=(A F)/(F B)+(A E)/(E C)dot