Question

State, without proofs all theorems of neutral geometry that permit to construct a midpoint of a given segment.

State, without proofs all theorems of Euclidean geometry that permit to execute your construction in the Poincare the Model of the Hyperbolic geometry.

Answer 1

Answer :-

Theorems of neutral geometry that permit to construct a midpoint of a given segment are:-

Definition 1:-  A point M is said to be a midpoint of a segment AB if A ∗ M ∗ B and AM ∼= MB.

Theorem 1:-  If A, B, C are points and A ∗ B ∗ C, then AB + BC = AC.

Theorem 2:- Given three distinct collinear points, exactly one of them lies between the other two.

Theorem 3:-  Every segment has a unique midpoint.

Theorems of Euclidean geometry that permit to execute your construction in the Poincare the Model of the Hyperbolic geometry are:-

Theorem 1:- The Poincar´e disk is a model of neutral geometry plus the hyperbolic parallel postulate.

And also these postulates:-

(I) For every line l and every point P not on l, there are at least two lines parallel to l and containing
P (Hyperbolic Parallel Postulate).
(II) If l and m are lines cut by a transversal t in such a way that the measures of two consecutive
interior angles adds up to less than 180◦
, then l and m need not necessarily intersect.
(III) Parallel lines are not everywhere equidistant.
(IV) If two parallel lines are cut by a transversal, the alternate interior angles need not be congruent.
(V) The sum of the measures of the three angles of any triangle is strictly less than 180°
(The Defectis Positive).
(VI) There exist triangles with different angle sums.
(VII) If two triangles are similar, then they are congruent.
(VIII) If two triangles have all three angles congruent, then the triangles are congruent. (AAA Congruence)
(IX) There is a universal upper bound to the area of any triangle. In other words, there exists a real
number K > 0 such that if 4ABC is any triangle, α(4ABC) ≤ K.