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)`

Let I be the incetre of Delta ABC having inradius r. Al, BI and Ci intersect incircle at D, E and F respectively. Prove that area of DeltaDEF " is " (r^(2))/(2) (cos.(A)/(2) + cos.(B)/(2) + cos.(C)/(2))

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)

A triangle A B C is inscribed in a circle with centre at O , The lines A O ,B Oa n dC O meet the opposite sides at D , E ,a n dF , respectively. Prove that 1/(A D)+1/(B E)+1/(C F)=(acosA+bcosB+ccosC)/

cos^2 A + cos^2 B +cos^2 C=1+2cosA cosB cosC .

Prove that cosA -cosB -cosC =1-4sin(A/2)cos(B/2)cos(C/2) ,if A+B+C= pi

For triangle ABC,R=5/2 and r=1. Let I be the incenter of the triangle and D,E and F be the feet of the perpendiculars from I->BC,CA and AB, respectively. The value of (IDXIEXIF)/(IAXIBXIC) is equal to (a) 5/2 (b) 5/4 (c) 10 (d) 1/5

Let A B C be a triangle with incentre at Idot Also, let Pa n dQ be the feet of perpendiculars from AtoB Ia n dC I , respectively. Then which of the following results are correct? (A P)/(B I)=(sinB/2cosC/2)/(sinA/2) (b) (A Q)/(C I)=(sinC/2cosB/2)/(sinA/2) (A P)/(B I)=(sinC/2cosB/2)/(sinA/2) (d) (A P)/(B I)+(a Q)/(C I)=sqrt(3)if/_A=60^0

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))

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 ExC F=A B^2dot Prove that the line E F always passes through a fixed point.

Let C be incircle of A B Cdot If the tangents of lengths t_1,t_2a n dt_3 are drawn inside the given triangle parallel to sidese a , ba n dc , respectively, the (t_1)/a+(t_2)/b+(t_3)/c is equal to 0 (b) 1 (c) 2 (d) 3