What is the intercept theorem (ratio theorem)?
The intercept theorem is about the ratio of line segments. We have two lines intersecting in point S. Two parallel lines (orange) intersect them in points A, B, C and D.
The points make up various line segments such as (the line from S to A), (the line from A to C) and so on. The theorem tells us about the ratios of the lengths of those line segments. That is useful if we know some of them but not all, then we can use the intercept theorem and solve it for the line segment that we want to calculate.
The ratio of two segments on one line are equal to the ratio of the corresponding two segments on the other line:
The ratio of two segments on one line are equal to the ratio of the parallel segments:
For more advanced uses of the intercept theorem, have a look at the Wikipedia article at http://en.wikipedia.org/wiki/Intercept_theorem.