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)))`

Explanation

To solve the problem, we will follow a structured approach to derive the required equation step by step.

### Step 1: Understand the Problem
We have a triangle \( ABC \) with incenter \( I \) and inradius \( r \). The points \( D, E, F \) are the feet of the perpendiculars from \( I \) to the sides \( BC, CA, \) and \( AB \), respectively. We need to prove the equation involving the inradii \( r_1, r_2, r_3 \) of the quadrilaterals \( AFIE, BDIF, \) and \( CEID \).

### Step 2: Establish Relationships
From the properties of the incenter and the quadrilaterals formed, we can relate the inradii \( r_1, r_2, r_3 \) to the angles of triangle \( ABC \):
- For quadrilateral \( AFIE \):
...