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` .

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 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

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

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 A ,B ,a n dC are three points on the hyperbola x y=c^2 such that A B subtends a right angle at C , then prove that A B is parallel to the normal to the hyperbola at point Cdot

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

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

In a triangle, if the angles A , B ,a n dC are in A.P. show that 2cos1/2(A-C)=(a+c)/(sqrt(a^2-a c+c^2))

Let the angles A , Ba n dC of triangle A B C be in AdotPdot and let b : c be sqrt(3):sqrt(2) . Find angle Adot