Solve Menelau's Theorem: Negative & Positive Lengths in a Plane

Thread starter disregardthat
Start date Feb 21, 2008
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SUMMARY

Menelau's Theorem states that if a transversal intersects the sides of a triangle, the product of the ratios of the segments created is equal to -1. The discussion highlights the confusion surrounding the distinction between positive and negative lengths in a plane, particularly when a point divides a segment internally or externally. The key takeaway is that internal division yields positive ratios, while external division results in negative ratios. This understanding clarifies the theorem's application and the reasoning behind the negative product.

PREREQUISITES

Understanding of Menelau's Theorem
Basic knowledge of triangle geometry
Familiarity with segment ratios
Concept of internal and external division of line segments

NEXT STEPS

Study the proof of Menelau's Theorem in detail
Explore the concept of internal vs. external division in geometry
Learn about the implications of negative ratios in geometric proofs
Investigate related theorems such as Ceva's Theorem

USEFUL FOR

Mathematicians, geometry students, and educators seeking to deepen their understanding of triangle properties and the application of Menelau's Theorem in various geometric contexts.

Feb 21, 2008

disregardthat

Science Advisor

[SOLVED] Menelau's theorem

"If a transversal is drawn to cut the sides of a triangle (produced if necessary), the product of the ratios of alternate segments is minus one.

So, for http://www.freewebs.com/jarle10/menelau's%20theorem.JPG (See picture)

[tex]\frac{BU}{UC} \cdot \frac{CY}{YA} \cdot \frac{AZ}{ZB} = -1[/tex]"

I don't understand this. How can they differ between negative and positive lengths in a plane? In the proof, an obvious step is that AY=-YA. But BZ apparently equal ZB. This makes it even more confusing. Can someone please explain how and why they differ between negative and positive values of the lengths in the plane?

I proved the case where the line does not intersect the triangle in a similar way as the first proof. The reason for that the product is minus one beats me. Why differ between positive and negative values?

Appreciate any help.

EDIT: I think I got it. When a point is internally dividing a line segment, the ratio is positive, and when it's dividing the segment extrernally, it's negative, as mentioned earlier in the chapter. yay

Last edited by a moderator: May 3, 2017