I How Does the Hyperbolic Axiom Define a Maximal Triangle?

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In hyperbolic geometry, a triangle can be defined with one side being a line R and two other lines through a point P not on R, which can be parallel to R. The concept of a "maximal triangle" is ambiguous and depends on the measure being considered, such as circumference, area, or angles. The discussion highlights that with parallel sides, the shape may not conform to traditional triangle definitions in Euclidean geometry. Clarification is needed on what "maximal" specifically refers to in this context. Understanding these principles requires familiarity with hyperbolic axioms and their implications for triangle properties.
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I know that if a triangle have it edge // to each orther then it í the maximum triangle.

Pls explain i don't understand hơ thí even possible
 
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Quantum Velocity said:
I know that if a triangle have it edge // to each orther then it í the maximum triangle.

Pls explain i don't understand hơ thí even possible
With parallel sides, it isn't a triangle anymore. Except you're not using Euclidean geometry. So a) where are the triangles defined on, and b) what does "maximal" mean: circumference, area, one angle, a side?
 
it's some type of geometry that i don't remember the name
 
it's called Lobachevshy-Bolyai geometry
 
or sometime called Hyperbolic geometry
 
You simply apply the hyperbolic axiom:

For any given line R and point P not on R, in the plane containing both line R and point P
there are at least two distinct lines through P that do not intersect R.


This is a given fact. So with two points on R, and a point P, not on R, we have three different points which define a triangle. And the two lines through P with the line R are parallel and the sides of the triangle. (Not quite sure about the sides.)

What maximal means, still depends on your measure.
 
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