Re: universal quantification, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

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