In ` A B C ,` the three bisectors of the angle A, B and C are extended to intersect the circumeircle at D,E and F respectively. Prove that `A DcosA/2+B EcosB/2+C FcosC/2=2R(sinA+sinB+sinC)`

Let I be the incentre of A B C having inradius rdotA I ,B Ia n dC I intersect incircle at D , Ea n dF respectively. Prove that area of D E F is (r^2)/2(cosA/2+cosB/2+cosC/2)

In A B C , the bisector of the angle A meets the side BC at D andthe circumscribed circle at E. Prove that D E=(a^2secA/2)/(2(b+c))

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

If A+B+C=pi , prove that : (cosA)/(sinb sinC) + (cosB)/(sinC sin) + (cosC)/(sinA sinB) =2 .

If A+B+C= pi/2 , show that : sin^2 A + sin^2 B + sin^2 C=1-2 sinA sinB sinC

If A ,Ba n dC are the vetices of a triangle A B C , then prove sine rule a/(sinA)=b/(sinB)=c/(sinC)dot

In A B C with fixed length of B C , the internal bisector of angle C meets the side A Ba tD and the circumcircle at E . The maximum value of C D.D E is c^2 (b) (c^2)/2 (c) (c^2)/4 (d) none of these

In acute angled triangle A B C ,A D is the altitude. Circle drawn with A D as its diameter cuts A Ba n dA Ca tPa n dQ , respectively. Length of P Q is equal to /(2R) (b) (a b c)/(4R^2) (c) 2RsinAsinBsinC (d) delta/R

Incrircle of A B C touches the sides BC, CA and AB at D, E and F, respectively. Let r_1 be the radius of incricel of B D Fdot Then prove that r_1=1/2(s(-b)sinB)/((1+sinB/2))