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.