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

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

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 .

If an a triangle A B C , b=3 c, a n d C-B=90^0, then find the value of tanB

If D , E ,a n dF are three points on the sides B C ,A C ,a n dA B of a triangle A B C such that A D ,B E ,a n dC F are concurrent, then show that BD*CE*AF=DC*EA*FB .

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.

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 A B C be a given isosceles triangle with A B=A C . Sides A Ba n dA C are extended up to Ea n dF , respectively, such that B E*C F=A B^2dot Prove that the line E F always passes through a fixed point.

In triangle A B C , prove that cos e c A/2+cos e c B/2+cos e c C/2geq6.

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