`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`

If D ,Ea n dF are three points on the sides B C ,C Aa n dA B , respectively, of a triangle A B C such that the (B D)/(C D),(C E)/(A E),(A F)/(B F)=-1

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 .

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 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

D , E ,a n dF are the middle points of the sides of the triangle A B C , then centroid of the triangle D E F is the same as that of A B C orthocenter of the tirangle D E F is the circumcentre of A B C orthocenter of the triangle D E F is the incenter of A B C centroid of the triangle D E F is not the same as that of A B Cdot

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

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

In a scalene triangle A B C ,D is a point on the side A B such that C D^2=A D D B , sin s in A S in B=sin^2C/2 then prove that CD is internal bisector of /_Cdot

Fid the condition if lines x=a y+b ,z=c y+da n dx=a^(prime)y+b^(prime), z=c^(prime)y+d ' are perpendicular.

Find the condition if lines x=a y+b ,z=c y+da n dx=a^(prime)y+b^(prime), z=c^(prime)y+d ' are perpendicular.