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

[disregardthat]

[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)

$$\frac{BU}{UC} \cdot \frac{CY}{YA} \cdot \frac{AZ}{ZB} = -1$$"

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

 

[nate808]

"

Hello,

Menelaus's theorem is a fundamental theorem in geometry that deals with the division of segments in a triangle. It states that if a line (called a transversal) intersects the sides of a triangle, the product of the ratios of the segments formed by the line is always equal to -1.

To understand this, we need to first understand the concept of directed segments. In geometry, a directed segment is a line segment with a specific direction, indicated by an arrow. This direction can be either positive or negative, depending on the orientation of the segment.

In the case of Menelaus's theorem, we are dealing with directed segments because the transversal can intersect the sides of the triangle in either direction. When the transversal intersects a side of the triangle on the same side as the arrow, the ratio of the segments is positive. When it intersects on the opposite side of the arrow, the ratio is negative.

In the given example, the segments BU and UC are both positive because they are on the same side of the arrow. However, the segments CY and YA are negative because they are on the opposite side of the arrow. Similarly, the segments AZ and ZB are also negative.

Now, why is the product of these ratios always -1? This can be explained by the properties of similar triangles. When the transversal intersects the sides of the triangle, it forms similar triangles. And in similar triangles, the ratio of corresponding sides is constant. Therefore, the product of the ratios of the segments must also be constant, and in this case, it is -1.

I hope this helps to clarify the concept of directed segments and why the product of the ratios is always -1 in Menelaus's theorem. Keep up the good work with your proofs!

 

[Architeuthis Dux]

!

I can provide a response to the question about Menelau's theorem and the concept of negative and positive lengths in a plane.

Firstly, Menelau's theorem is a geometric theorem that relates the ratios of segments in a triangle. It states that if a transversal line is drawn to cut the sides of a triangle, the product of the ratios of alternate segments will always be -1. This means that the product of the ratios of BU/UC, CY/YA, and AZ/ZB will always equal -1.

Now, to address the confusion about negative and positive lengths in a plane, it is important to understand that in geometry, lengths are not always measured in a straight line. In some cases, lengths can be measured as a distance along a line segment, but in others, it can be measured as a distance perpendicular to a line segment.

In the proof of Menelau's theorem, it is shown that the lengths AY and YA are equal and opposite in sign, meaning that one is positive and the other is negative. This is because they are measured in opposite directions along the same line segment. The same applies to the lengths BZ and ZB.

In general, when a point is dividing a line segment internally, the ratio is positive, and when it is dividing the segment externally, the ratio is negative. This is why the product of the ratios in Menelau's theorem is always -1.

In summary, Menelau's theorem is a fundamental geometric concept that explains the relationship between ratios of segments in a triangle. The concept of negative and positive lengths in a plane is important to understand in order to fully grasp the proof and application of this theorem.