Similar triangle - naming issue in Euclid book

Mondo

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Hi,

I read proposition 8 in Euclid elements book:



In its punch line it says "So I say the triangles ABD and ADC are also similar to one another" - Ain't the order of angles important? I mean when they say `ABD` is similar to `ADC` shouldn't they really care about the angle to be the same? So instead of `ABD` similar to `ADC` they really should say `ABD` is similar to `DAC` since these angles are equal? I cross checked it with this excellent internet reference https://mathcs.clarku.edu/~djoyce/elements/bookVI/propVI8.html and here it is all correct. What do you guys think?
 

lev888

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Hi,

I read proposition 8 in Euclid elements book:



In its punch line it says "So I say the triangles ABD and ADC are also similar to one another" - Ain't the order of angles important? I mean when they say `ABD` is similar to `ADC` shouldn't they really care about the angle to be the same? So instead of `ABD` similar to `ADC` they really should say `ABD` is similar to `DAC` since these angles are equal? I cross checked it with this excellent internet reference https://mathcs.clarku.edu/~djoyce/elements/bookVI/propVI8.html and here it is all correct. What do you guys think?
I've never heard about this requirement. If one triangle is similar to two triangles, would you refer to the first triangle using 2 different names?
 

Dr.Peterson

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In its punch line it says "So I say the triangles ABD and ADC are also similar to one another" - Ain't the order of angles important? I mean when they say `ABD` is similar to `ADC` shouldn't they really care about the angle to be the same? So instead of `ABD` similar to `ADC` they really should say `ABD` is similar to `DAC` since these angles are equal? I cross checked it with this excellent internet reference https://mathcs.clarku.edu/~djoyce/elements/bookVI/propVI8.html and here it is all correct. What do you guys think?
Your source is different in detail from Joyce's version; the latter says at the end, "Therefore the triangle DBA is similar to the triangle DAC." This is correct; your "`ABD` is similar to `DAC`" is as wrong as "`ABD` is similar to `ADC`".

So either your source is more directly translated from Euclid, who may well not yet have used the modern convention of naming figures in corresponding order, or else Joyce has it correct and your source was not careful about that. I suspect the former is more likely, in light of Joyce's comments on the definition, https://mathcs.clarku.edu/~djoyce/elements/bookVI/defVI1.html

You can see the actual Greek here, https://www.claymath.org/library/historical/euclid/files/elem.6.8.html, which has (in its translation) "therefore the triangle ABD is similar to the triangle ADC". This supports the idea that Euclid did not follow the modern convention. I haven't searched for further confirmation of that.
 

lex

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I mean when they say ABD is similar to ADC shouldn't they really care about the angle to be the same?
No, they say 'triangle ABD', which is a reference to a triangle, not to an angle ('angle ABD').

There's no problem with the naming. ADC is just a name for the triangle. ACD refers to the same triangle. It is the triangles that are similar shapes and it is that information which is being communicated. The precise detail of how the triangles are referred to doesn't matter. When I say \(\displaystyle \bigtriangleup\) ADC is similar to \(\displaystyle \bigtriangleup\) ABD, I am saying that the set of three angles in the triangle that the label '\(\displaystyle \bigtriangleup\) ADC' refers to, is the same as the set of three angles in the triangle that the label '\(\displaystyle \bigtriangleup\) ABD' refers to. I am not saying anything directly about \(\displaystyle \angle ADC\) as compared to \(\displaystyle \angle ABD\).
In fact he could have conveyed the same mathematical information by referring to the triangles differently e.g. the two small triangles are similar to the biggest triangle.
(There may be something slightly satisfying about having them named in a way to suggest which angles are the same, but as this is by no means necessary and certainly cannot be relied upon, I wouldn't give it much credence. I wouldn't give a moment's thought to the order. I would just name the triangles in a 'legal' way).
 

Dr.Peterson

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(There may be something slightly satisfying about having them named in a way to suggest which angles are the same, but as this is by no means necessary and certainly cannot be relied upon, I wouldn't give it much credence. I wouldn't give a moment's thought to the order. I would just name the triangles in a 'legal' way).
We need to clarify something.

@Mondo is asking about a common convention by which a statement about congruence or similarity is taken to imply correspondences by naming the figures in corresponding orders. For example, in https://www.cliffsnotes.com/study-guides/geometry/triangles/congruent-triangles we are told, "Congruent triangles are named by listing their vertices in corresponding orders. In Figure , Δ BAT ≅ Δ ICE."

You're right that it is not necessary; but it is not nonexistent. It is very much worth being aware of.

As I said, "Euclid ... may well not yet have used the modern convention of naming figures in corresponding order." But it is very commonly assumed today -- in fact, I had a hard time finding an explicit statement of this because when writers refer to corresponding parts, they tend to assume the convention without even mentioning it (especially in the context of similarity, which is taught after students have become accustomed to the convention for congruence). I find it being used without mention all over the place, wherever someone states a congruence and then lists corresponding parts without questioning them.

I would definitely give more than a moment's thought to the order! Without the convention, you have to figure out from context which parts correspond, so it is very useful when adhered to (and therefore confusing when not). Therefore, I would want to determine whether an author is using it or not, and then read with that knowledge. That is what we are discussing.
 

lex

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@Dr.Peterson
No, the issue raised, which I have addressed, was whether it was correct.

(On the issue of a convention that you have brought up, I have managed never to come across it and clearly others are in a similar position - lev888, and JeffM #2).
 

Mondo

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Ok I think it is not required and even not possible since as lev888 said it can't be done for more than two similar triangles. However I think its better to take care of the angle when you have only two triangles as there is no doubt plus you quickly get the common angle.
 
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Dr.Peterson

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Ok I think it is not required and even not possible since as lev888 said it can't be done for more than two similar triangles. However I think its better to take care of the angle when you have only two triangles as there is no doubt plus you quickly get the common angle.
It's entirely possible with more than two triangles. In this case, \(\displaystyle \triangle ABD \sim \triangle CAD \sim \triangle CBA\). All corresponding vertices are indicated here.

May I ask why you asked your question? You said, "Ain't the order of angles important? I mean when they say ABD is similar to ADC shouldn't they really care about the angle to be the same? So instead of ABD similar to ADC they really should say ABD is similar to DAC since these angles are equal?" That implies that you have been taught this convention, and thought it was required. That is what I have been answering.

I just want to be sure I am properly answering your question, in your context. I've already said that Euclid was not incorrect. I'm not the only person here who has heard of the convention, right?
 

JeffM

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I may have heard about it back in 1960 when I formally studied plane geometry, but I certainly never remembered it. I was impressed by lev's comment, which implies such a naming convention cannot be a requirement. But I certainly can see how it improves communication in certain circumstances, and I shall try to keep it in mind in the future.
 

lev888

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I may have heard about it back in 1960 when I formally studied plane geometry, but I certainly never remembered it. I was impressed by lev's comment, which implies such a naming convention cannot be a requirement. But I certainly can see how it improves communication in certain circumstances, and I shall try to keep it in mind in the future.
On second thought, there is no need to rename the common triangle - the other 2 can be renamed accordingly.
 

JeffM

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Let me try again. Naming is fundamentally arbitrary. Nevertheless, some naming conventions are more useful in facilitating communication than others. So I shall try to do better without becoming perfectionist or condemnatory. Which I think was Dr. Peterson’s point all along.
 

lex

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Let me try again. Naming is fundamentally arbitrary. Nevertheless, some naming conventions are more useful in facilitating communication than others. So I shall try to do better without becoming perfectionist or condemnatory. Which I think was Dr. Peterson’s point all along.
I applaud your optimism and generosity of spirit.
 

Mondo

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It's entirely possible with more than two triangles. In this case, \(\displaystyle \triangle ABD \sim \triangle CAD \sim \triangle CBA\). All corresponding
Yeah I was wrong, we certainly can reference each similar triangles using common angles and yes, this is how I was taught. And this copy of Euclid https://mathcs.clarku.edu/~djoyce/elements/bookVI/propVI8.html follows these rules while the book doesn't. Even if this convention is not necessary I think its just better, as we discussed and agreed. Thanks!
 
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