# Euclid's elements proposition 15 book 3

• astrololo
In summary, the conversation discusses a small detail in a mathematical proof regarding the relationship between perpendiculars and straight lines. The participant is unsure about the validity of the proof as it seems to use the inverse of a definition. Another participant clarifies that all definitions are "if and only if," meaning the inverse is also true. Therefore, the proof is valid.
astrololo
http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html

I have understood the proof in general. It is only a small detail which I'm not sure. Maybe it's because english isn't my first language. Anyway, the part of the proof which says : "Then, since BC is nearer to the center and FG more remote, EK is greater than EH." I have no problem understanding what is said here. This is supported by this definition : "And that straight line is said to be at a greater distance on which the greater perpendicular falls." (Definition 5 of book 3) Now, this is where I'm unsure. From what I understand of it, it says that if I have a perpendicular that is bigger than the other, than my straight line is said to be at a greater distance. (This is how I understand it) Now, in the proof, we do the inverse. We know that one line is at a greater distance than the other and we conclude with the definition that one perpendicular is bigger than the other. How is this correct ? Unless the definition implies that the reverse is also ok, then this works. But if the definition implies only one direction, (The one which is defined) then how is the proof valid ?

By the way, you don't need to read all of the proof. Only the things at beginning are needed.

All definitions are "if and only if". To say that the "that straight line is said to be at a greater distance on which the greater perpendicular falls." is the same as saying "a straight line is at a greater distance if and only if it has the greater perpendicular."

Yes I understand this. You mean that "if and only if" makes the inverse(converse) also true, right ?

## What is Euclid's Elements Proposition 15 Book 3?

Euclid's Elements is a mathematical treatise written by the ancient Greek mathematician Euclid around 300 BC. It is divided into 13 books, each containing a set of propositions and their corresponding proofs. Proposition 15 in Book 3 deals with the properties of parallelograms.

## What is the statement of Proposition 15 Book 3?

Proposition 15 in Book 3 states that if a line is drawn parallel to one side of a parallelogram, it will divide the other two sides proportionally. This is also known as the parallelogram law or the midpoint theorem.

## What is the significance of Proposition 15 Book 3?

Proposition 15 in Book 3 is significant because it is one of the fundamental theorems in the study of parallelograms and their properties. It is also used as a building block for other geometric proofs and theorems.

## How did Euclid prove Proposition 15 Book 3?

Euclid used a combination of axioms, definitions, and previously proven propositions to arrive at the proof for Proposition 15 in Book 3. The proof involves constructing similar triangles and using the properties of parallel lines.

## How is Proposition 15 Book 3 applicable in real life?

The parallelogram law or midpoint theorem has applications in various fields such as engineering, architecture, and physics. It is used to find the center of mass, calculate forces and moments in structures, and determine the stability of an object.

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