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

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

Let the base A B of a triangle A B C be fixed and the vertex C lies on a fixed circle of radius rdot Lines through Aa n dB are drawn to intersect C Ba n dC A , respectively, at Ea n dF such that C E: E B=1:2a n dC F : F A=1:2 . If the point of intersection P of these lines lies on the median through A B for all positions of A B , then the locus of P is (a) a circle of radius r/2 (b) a circle of radius 2r (c) a parabola of latus rectum 4r (d) a rectangular hyperbola

In triangle A B C ,points D , Ea n dF are taken on the sides B C ,C Aa n dA B , respectively, such that (B D)/(D C)=(C E)/(E A)=(A F)/(F B)=ndot Prove that triangle D E F=(n^2-n+1)/((n+1)^2) triangle(A B C)dot

If the vectors vec a , vec b ,a n d vec c form the sides B C ,C Aa n dA B , respectively, of triangle A B C ,t h e n

If Aa n dB are two independent events such that P(A)=1//2 a n d P(B)=1//5, then

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

If A B C D is quadrilateral and Ea n dF are the mid-points of A Ca n dB D respectively, prove that vec A B+ vec A D + vec C B + vec C D =4 vec E Fdot

If A(-2,1),B(2,3)a n dC(-2,-4) are three points, find the angle between B Aa n dB Cdot

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .