Equal intercept theorem

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.

what is an intercept

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.

what is equal intercept theorem

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

Proof of equal intercept theorem

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

Join points PL which meet line B at point O

To prove:
The intercept formed by line N are equal.
i.e. RQ = QL

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.

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