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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 13 May 2024 13:43:01 -0700
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On 05/13/2024 01:22 PM, Ross Finlayson wrote:
> On 05/12/2024 09:11 PM, Jim Burns wrote:
>> On 5/12/2024 3:34 PM, Ross Finlayson wrote:
>>> On 05/12/2024 11:21 AM, Ross Finlayson wrote:
>>>> On 05/12/2024 10:46 AM, Jim Burns wrote:
>>
>>>>> [...]
>>>>
>>>> Geometry's "similar" is often related to
>>>> "congruency, thoroughly",
>>>> where "congruent" means "similar".
>>
>> You have confused me.
>> I agree that "similar" is related to "similar"
>> but I don't see why you tell us this.
>>
>>>>>   ͨₐ🭢🭕🭞🭜🭘ᵇ  =  ͨₐ🭢🭕ͩ   +  ͩₐ🭞🭜🭘ᵇ
>>>>>
>>>>> The right triangle 🞃cab is split into
>>>>> two right triangles ◥cda ◤adb
>>>>> by segment a͞d perpendicular to b͞c
>>>>>
>>>>> 🞃cab ◥cda ◤adb are _similar_
>>>>> [1]
>>>>> Corresponding sides have equal ratios.
>>
>> Because triangles, here "similar" means
>> corresponding sides have equal ratios and
>> corresponding angles are equal.
>>
>>>> You can know that a triangle is a right triangle
>>>> if you have the trigonometric functions of its angles,
>>>> here as where it doesn't necessarily require
>>>> the apparatus of Pythagoren theorem proper,
>>>> "its own theory", ....
>>
>> Pythagoras says
>> ∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
>>
>> That means
>> ∠cab = 90°  ∨  c͡b² ≠ c͡a² + b͡a²
>>
>> One side of that disjunction is true
>> for any triangle.
>> Thus, we don't need to know it's a right triangle
>> in order to know Pythagoras is correct.
>>
>>> The most usual tools, of classical constructions:
>>> are: compass and edge.
>>
>> If 🞃cab CAN be classically constructed
>> then  ∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
>>
>> If 🞃cab canNOT be classically constructed
>> then  ∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
>>
>>
>
> I just showed there's another way to arrive at the objects
> of the functions of trigonometry, independent of the
> usual way. Now, the usual way what was a uniqueness
> result, is now a distinctness result, what was a singularity,
> is now a multiplicity, and what had no end, now has two.
>
> For any planar triangle, that any two of its angles
> sum to a right angle, indicates the triangle is right.
>
> So, in this context of universal quantification, you had
> brought up the point about the Pythagorean theorem
> as if it was only available as a definition, and now I just
> showed that it's also as after a derivation, and vice-versa.
>
> Then as with regards to their conflation, and combination,
> their confluence, "right" and "equilateral" trigonometry,
> makes for as simple an example as you presented that
> a deconstructive and reconstructive account provides a
> greater dialectic and reasoning grounds for disambiguation.
>
>
> And Zeno's like "you know these universal quantifiers can
> help if you can explain how they neatly encapsulate an
> expressed intent the actual vis-a-vis the potential infinity".
>
>
> No, I don't know where else such ideas about this novel
> "equilateral trigonometry" are. Here though there is one.
>
>

Also triangles that are very narrow and very tall, and look right.