`A B C D` is a tetrahedron and `O` is any point. If the lines joining `O` to the vertices meet the opposite faces at `P ,Q ,Ra n dS ,` prove that `(O P)/(A P)+(O Q)/(B Q)+(O R)/(C R)+(O S)/(D S)=1.`

Explanation

To prove the equation \(\frac{OP}{AP} + \frac{OQ}{BQ} + \frac{OR}{CR} + \frac{OS}{DS} = 1\) for the tetrahedron \(ABCD\) with point \(O\) and points \(P, Q, R, S\) being the intersections of lines from \(O\) to the vertices with the opposite faces, we can follow these steps:

### Step 1: Define Position Vectors
Let the position vectors of points \(A, B, C, D\) be represented as \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) respectively. Let the position vector of point \(O\) be \(\vec{o}\).

### Step 2: Express Points \(P, Q, R, S\)
The points \(P, Q, R, S\) are defined as the intersections of the lines \(OP, OQ, OR, OS\) with the opposite faces. We can express these points in terms of the position vectors of \(O\) and the vertices of the tetrahedron.

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