If `D ,E and F` are three points on the sides `B C ,CA and A B ,` respectively, of a triangle `A B C` show that the `(B D)/(C D)=(C E)/(A E)=(A F)/(B F)=-1`

Explanation

To prove that \(\frac{BD}{CD} = \frac{CE}{AE} = \frac{AF}{BF} = -1\), we start by considering the triangle \(ABC\) with points \(D\), \(E\), and \(F\) on sides \(BC\), \(CA\), and \(AB\) respectively.

### Step 1: Assign Position Vectors
Let the position vectors of points \(A\), \(B\), and \(C\) be represented as \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\). We can express the position vectors of points \(D\), \(E\), and \(F\) in terms of these vectors.

### Step 2: Express the Position Vectors
Assume:
- Point \(D\) divides \(BC\) in the ratio \(k:1\), so we can write:
...