In this chapter we will understand and prove the equal intercept theorem.

Before moving on to learn the theorem, let us first understand the basics.

**What is an intercept ?**

When a transversal line intersect the two given lines then the length of transversal between the given two lines is called an intercept.

For example, consider the below figure.

Here M & N are two parallel lines intersected by transversal P.

The length of segment AB is called an intercept.

## Equal intercept theorem

According to the theorem, **if a transversal form equal intercept by intersecting three or more parallel lines then another transversal will also form equal intercept**.

Given above is the parallel lines A, B and C intersecting by transversal M & N.

Here the intercept formed by transversal M are equal.

i.e. PS = ST

According to the equal intercept theorem,** if another transversal intersect the same parallel lines then the intercept formed will also be equal**.

Here line N intersect the same three parallel lines, so the intercept formed are also equal.

i.e. **RQ = QL**

### Proof of equal intercept theorem

**Given:**

Given above are parallel lines A, B and C intersected by transversals M & N.

The intercepts formed by line M are equal.

i.e. PS = ST**Construction:**

Join points PL which meet line B at point O**To prove:**

The intercept formed by line N are equal.

i.e. RQ = QL

**Proof:****Consider triangle PTL.**

We know that;

PS = ST { given }

SO is parallel to TL;

Then according to midpoint theorem of triangle, O is the midpoint of side PL.

i.e. PO = OL

**Now consider triangle LPR;**

We know that;

OQ is parallel to PR { given }

PO = OL { Proved above }

Again, using the midpoint theorem we can say that ” Q is the midpoint of RL “

i.e. RQ = QL

Hence, **we proved the equal intercept theorem**.