Let ABC be a triangle with incenter I and inradius r. Let D, E, and F be the feet of the perpendiculars from I to the sides BC, CA and AB, respecitvely. If `r_(1), r_(2), and r_(3)` are the radii of circles inscribed in the quadrilaterals AFIE, BDIF, and CEID, respectively, prove that

Let ABC be a triangle with incentre I and inradius r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB, respectively, If r_(2)" and "r_(3) are the radii of circles inscribed in the quadrilaterls AFIE, BDIF and CEID respectively, then prove that r_(1)/(r-r_(1))+r_(2)/(r-r_(2))+r_(3)/(r-r_(3))=(r_(1)r_(2)r_(3))/((r-r_(1))(r-r_(2))(r-r_(3)))

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 ABC be a triangle with incentre I. If P and Q are the feet of the perpendiculars from A to BI and CI, respectively, then prove that (AP)/(BI) + (AQ)/(Cl) = cot.(A)/(2)

For a triangle ABC, R = (5)/(2) and r = 1 . Let D, E and F be the feet of the perpendiculars from incentre I to BC, CA and AB, respectively. Then the value of ((IA) (IB)(IC))/((ID)(IE)(IF)) is equal to _____

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: area of BDEF is half the area of Delta ABC.

In a triangle ABC if sides a = 13, b =14 and c = 15, then reciprocals of r_(1),r_(2),r_(3) are in the ratio

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 (ID*IE*IF)/(IA*IB*IC) is equal to (a) 5/2 (b) 5/4 (c) 1/10 (d) 1/5

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: BDEF is a parallelogram.

In Delta ABC , if r_(1) lt r_(2) lt r_(3) , then find the order of lengths of the sides

D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral Delta ABC. Show that Delta DEF is also an equilateral triangle.