Weird configuration of the Desargues' theorem. Is it correct?

The theorem states that two triangles are in perspective axially if and only if they are in perspective centrally. In summary, the given diagram is a correct representation of Desargues' theorem.
  • #1
Iron-ben
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0
Desargues' theorem:
Two triangles are in perspective axially if and only if they are in perspective centrally.

I was wondering if this particular configuration was correct:

http://garciacapitan.99on.com/bella/imgs/desarg2.gif

http://garciacapitan.99on.com/bella/imgs/desarg2.gif (copy-paste the link ir it doesn't show you the image or doble click on the attached thumbnail)

I don't know if it is even possible for the center of perspectivity (O) be in-between the two triangles. Also I believe that the perspectivity axis (P,Q,R) isn't horizontal.

My first thought was that the correspondent vertices were wrongly located but then again I found this in a geometry book. So please could you help me? Are the correspondent vertices wrongly located or am I not undertanding? HELP please!
 

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  • #2
Yes, the triangles in the diagram are in perspective axially, and the configuration is correct. The center of perspectivity (O) is located in-between the two triangles, and the perspectivity axis (P,Q,R) is not horizontal. The correspondent vertices of the two triangles are correctly located.
 

1. How does Desargues' theorem relate to geometry?

Desargues' theorem is a fundamental theorem in projective geometry that states that if two triangles are perspective from a point, then they are perspective from a line as well. This theorem is used to prove many other theorems in geometry.

2. What is the significance of the "weird configuration" in the Desargues' theorem?

The "weird configuration" in the Desargues' theorem refers to a situation where the two triangles are not in the usual position of being aligned with each other, but instead are in a more complex and seemingly "weird" arrangement. This configuration is important because it shows that the theorem holds true even in unusual cases.

3. How do you know if the "weird configuration" of the Desargues' theorem is correct?

To determine if the "weird configuration" of the Desargues' theorem is correct, one must carefully analyze the given triangles and their corresponding points and lines. The theorem can be proven using mathematical proofs and geometric constructions.

4. Are there any real-life applications of the Desargues' theorem and its "weird configuration"?

Yes, the Desargues' theorem and its "weird configuration" have various real-life applications, especially in computer graphics and computer-aided design. This theorem is also used in the design of perspective drawings and camera lenses.

5. Can the "weird configuration" of the Desargues' theorem be extended to higher dimensions?

Yes, the Desargues' theorem can be extended to higher dimensions, such as 3D space. In this case, it is known as the "Desargues' configuration" and involves two sets of three mutually perspective triangles. This higher-dimensional version of the theorem is important in the study of projective geometry and its applications.

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