`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 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 .

Let D be the middle point of the side B C of a triangle A B Cdot If the triangle A D C is equilateral, then a^2: b^2: c^2 is equal to 1:4:3 (b) 4:1:3 (c) 4:3:1 (d) 3:4:1

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 .

AD, BE and CF are the medians of triangle ABC whose centroid is G. If the points A, F, G and E are concyclic, then prove that 2a^(2) = b^(2) + c^(2)

Let P be a point interior to the acute triangle A B Cdot If P A+P B+P C is a null vector, then w.r.t triangle A B C , point P is its a. centroid b. orthocentre c. incentre d. circumcentre

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

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

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90^@-1/2A , 90^@-1/2B and 90^@-1/2C

In triangle A B C , base B C and area of triangle are fixed. The locus of the centroid of triangle A B C is a straight line that is a) parallel to side B C (b)right bisector of side BC (c)perpendicular to BC (d)inclined at an angle sin^(-1)((sqrt())/(B C)) to side BC