Intercept Theorem: Counter-Example & Proof

[lwymarie]

intercept theorem--urgent

does the converse to the intercept theorem hold? i.e. if three lines make equal intercepts on each of two transversals PQ and RS as shown, is it true that the three lines must be parallel? can you produce a counter-example? (a drawing indicated with length measurements in necessary)

I can draw the diagram of a counter-example but i don't know how to write the steps of proving it:(

 

[Andrew Mason]

does the converse to the intercept theorem hold? i.e. if three lines make equal intercepts on each of two transversals PQ and RS as shown, is it true that the three lines must be parallel? can you produce a counter-example? (a drawing indicated with length measurements in necessary)

I can draw the diagram of a counter-example but i don't know how to write the steps of proving it:(

Perhaps you could attach the drawing. The description of the problem is not clear.

AM

 

[thepatientmental]

Yes, the converse to the intercept theorem does hold. This can be proven using a counter-example.

Counter-example: Let's consider three lines, AB, CD, and EF, intersecting two transversals PQ and RS as shown in the diagram below.

<img src="" width="200">

We can see that AB, CD, and EF all make equal intercepts on both transversals PQ and RS. However, these three lines are not parallel.

Proof: To prove that the converse to the intercept theorem holds, we need to show that if three lines make equal intercepts on each of two transversals, then they must be parallel.

Let's label the points where the lines intersect the transversals as A, B, C, D, E, and F as shown in the diagram.

<img src="" width="200">

Since the intercept theorem states that the ratio of the intercepts is equal to the ratio of the intercepted segments, we can write the following equations:

AB/CD = PQ/RS
CD/EF = RS/PQ
EF/AB = PQ/RS

From these equations, we can see that AB/CD = EF/AB, which means that AB^2 = CD x EF.

Similarly, we can also see that CD/EF = AB/CD, which means that CD^2 = EF x AB.

Therefore, AB^2 = CD^2, which implies that AB = CD.

Similarly, we can also show that CD = EF and AB = EF.

This means that all three lines are equal in length, which is only possible if they are parallel.

Hence, we have proven that if three lines make equal intercepts on each of two transversals, they must be parallel, which is the converse to the intercept theorem.