Triangles which are on the same base and in the same paralle

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Discussion Overview

The discussion revolves around Proposition 37 of Euclid's Elements, specifically addressing the definition and properties of parallelograms. Participants explore whether the proof adequately demonstrates the existence of a parallelogram based on the given conditions of parallel sides.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether having a figure with four sides and opposite sides that are parallel is sufficient to conclude that it is a parallelogram.
  • Another participant asserts that this is indeed how a parallelogram is defined.
  • A subsequent reply raises the possibility that the figure could also be a rectangle, which is a specific type of parallelogram.
  • Further clarification is provided that a rectangle has all four interior angles equal, distinguishing it from other types of parallelograms.
  • Another participant seeks to understand if the discussion is referring to a general parallelogram, rather than a specific case, and acknowledges a prior misunderstanding regarding the term "rhomboid."
  • One participant corrects a previous statement about the nature of parallelograms, emphasizing that they typically have two pairs of parallel sides.

Areas of Agreement / Disagreement

Participants express differing views on whether the proof sufficiently establishes the existence of a parallelogram, and there is no consensus on the implications of the definitions discussed.

Contextual Notes

There are unresolved assumptions regarding the definitions of parallelograms and rectangles, as well as the implications of the proof in question.

astrololo
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I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html

The only problem I got with the proof is the fact that we don't seem to prove that we do have parallelogram. We have a figure with 4 side and we know that each opposite sides are parallels to each other. Is this considered enough to acknowledge that we have parallelograms ?

Thank you!
 
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astrololo said:
I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html

The only problem I got with the proof is the fact that we don't seem to prove that we do have parallelogram. We have a figure with 4 side and we know that each opposite sides are parallels to each other. Is this considered enough to acknowledge that we have parallelograms ?
That's how a parallelograph is defined.
 
Mark44 said:
That's how a parallelograph is defined.
But couldn't this be a rectangle ?
 
A rectangle is one type of parallelogram, one in which all four interior angles are equal.
 
Mark44 said:
A rectangle is one type of parallelogram, one in which all four interior angles are equal.
Ok, then he's not pointing to any specific case, just that we have a general parallelogram ? (That usually have 2 pairs of parallels) Is that how I'm supposed to understand it ? I did the mistake of assuming that we had a
Rhomboid
 
astrololo said:
Ok, then he's not pointing to any specific case, just that we have a general parallelogram ? (That usually have 2 pairs of parallels)
Fixed the above for you: "That usually always have 2 pairs of parallels"
astrololo said:
Is that how I'm supposed to understand it ? I did the mistake of assuming that we had a
Rhomboid
 
Mark44 said:
Fixed the above for you: "That usually always have 2 pairs of parallels"
thank you !
 

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