`O` is any point inside a triangle `A B C` . The bisector of `/_A O B ,/_B O C` and `/_C O A` meet the sides `A B ,B C` and `C A` in point `D ,Ea n dF` respectively. Show that `A D x B E x C F=D B x E C x F A`

Explanation

To prove that \( AD \times BE \times CF = DB \times EC \times FA \), we will use the angle bisector theorem in triangle geometry. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have triangle \( ABC \) with point \( O \) inside it. The angle bisectors of angles \( AOB \), \( BOC \), and \( COA \) meet sides \( AB \), \( BC \), and \( CA \) at points \( D \), \( E \), and \( F \), respectively.

### Step 2: Apply the Angle Bisector Theorem
According to the angle bisector theorem:
1. In triangle \( AOB \), since \( OD \) is the angle bisector of \( \angle AOB \):
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