Intercept Theorem

The Intercept Theorem is a fundamental tool of Euclidean Geometry. The concept of parallel lines and transversal is of great importance in our day-to-day life. And, the Intercept theorem extends our understanding of parallel lines and transversal and we can apply these concepts in our day-to-day life.

A Transversal

In the above figure, we can see that there are 3 parallel lines ${L}_{1}$,${L}_{2}$, ${L}_{3}$ and then there is a transversal $PR$ which is intersecting all the 3 parallel lines at an equal distance. The intercept theorem, also known as Thale’s theorem, Basic Proportionality Theorem, or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels.

History of the Mathematician

Euclid

• Year of Birth: 325 BC

• Year of Death: 270 BC

• Contribution: He contributed significantly in the field of Mathematics and Physics by discovering the intercept theorem.

Statement of the Theorem

If there are three or more parallel lines and the intercepts made by them on one transversal are equal, the corresponding intercepts of any transversal are also equal.

Proof of the Theorem

Two Parallel Lines

Given:

$l$, $m$, $n$ are three parallel lines.

$P$ is a transversal intersecting the parallel lines such that $AB=BC$.

The transversal $Q$ has the intercepts $DE$ and $HE$ by the parallel lines $l$, $m$, $n$.

To prove:

$DE=EF$

Proof:

Draw a line $E$ parallel to the line $P$ which intersects the line $n$ at $H$ and line $l$ at $G$.

$AG||BE$ (Given)

$GE||AB$ (By construction)

From the information above, we can say that $AGBE$ is a parallelogram.

According to the properties of a parallelogram:

$AB=GE$ - (1)

Similarly, we can say that $BEHC$ is a parallelogram.

$BC=HE$ - (2)

From the given information, we know that $AB=BC$.

So, from equations (1) and (2), we can say that $GE=HE$.

In $\Delta GED$ and $\Delta HEF$,

$GE=HE$(Proved)

$\angle GED=\angle FEH$(Vertically Opposite Angles)

$\angle DGE=\angle FHE$(Alternate Interior Angles)

Hence, $\Delta GED\cong \Delta HEF$

As$\Delta GED\cong \Delta HEF$, the sides$DE=EF$.

Hence proved.

Applications of the Theorem

The intercept theorem can be used to prove that a certain construction yields parallel line segments:

• If the midpoints of two triangle sides are connected, then the resulting line segment is parallel to the third triangle side (Mid point theorem of triangles).

• If the midpoints of the two non-parallel sides of a trapezoid are connected, then the resulting line segment is parallel to the other two sides of the trapezoid.

Limitations of the Theorem

• The intercept theorem is not able to help us in finding the midpoint of the sides of the triangle.

• The basic proportionality theorem is an advanced version of the intercept theorem and it gives us a lot of information on the sides of the triangles.

Solved Examples

1. In a \[\Delta ABC\], sides \[AB\] and \[AC\] are intersected by a line at \[D\] and \[E\], respectively, which is parallel to side \[BC\]. Prove that \[\dfrac{AD}{AB}=\dfrac{AE}{AC}\].

Ans:

Scalene Triangle

\[DE||BC\] (Given)

So, \[\dfrac{AD}{DB}=\dfrac{AE}{EC}\]

Interchanging the ratios,

\[\dfrac{DB}{AD}=\dfrac{EC}{AE}\]

Adding 1 to both sides,

\[\dfrac{DB}{AD}+1=\dfrac{EC}{AE}+1\]

\[\dfrac{AD+DB}{AD}=\dfrac{EC+AE}{AE}\]

\[\dfrac{AB}{AD}=\dfrac{AC}{AE}\]

Interchanging the ratios again,

\[\dfrac{AD}{AB}=\dfrac{AE}{AC}\]

Hence proved.

2. Find DE

Basic Proportionality Theorem

Ans: According to the basic proportionality theorem,

\[\dfrac{AE}{DE}=\dfrac{BE}{CE}\]

\[\dfrac{4}{DE}=\dfrac{6}{8.5}\]

\[\dfrac{4*8.5}{6}=DE\]

\[DE=5.66\]

So, \[DE=5.66\]

3. In \[\Delta ABC\], \[D\] and \[E\] are points on the sides \[AB\] and \[AC\], respectively, such that \[DE||BC\]. If \[\dfrac{AD}{DB}=\dfrac{3}{4}\] and \[AC=15cm\], find \[AE\].

Intercept Theorem

Ans:\[\dfrac{AD}{DB}=\dfrac{AE}{EC}\] (According to the intercept theorem)

Let \[AE=x\] and \[EC=15-x\]

\[\dfrac{AD}{DB}=\dfrac{3}{4}\] (Given)

So, \[\dfrac{3}{4}=\dfrac{x}{15-x}\]

\[3(15-x)=4x\]

\[45=7x\]

\[x=\dfrac{45}{7}\]

\[x=6.4cm\]

So, \[x=6.4cm\]

Important Points

• The intercept theorem can only be applied when the lies are parallel, if the transversal is cutting lines that are not parallel, then the intercept theorem is not valid.

• The basic proportionality theorem and mid-point theorem are all applications of the intercept theorem but they are not the same theorems.

Conclusion

In the above article, we have discussed the Equal intercept Theorem and its proof. We have also discussed the applications of the theorem. So, we can conclude that Intercept Theorem is a fundamental tool of Geometry and is based on applications of parallel lines and transversal and reduces our computational work based on its application as we have seen in the examples based on the theorem.