Download Non-Euclidean Geometry, spring term 2017 Homework 1. Due date

Non-Euclidean Geometry, spring term 2017
Homework 1. Due date: February 13.
Faculty of Mathematics, NRU HSE, and Math in Moscow, IUM
Write your solutions neatly or type in TeX. Late homeworks are not accepted.
Problem 1. Every segment in a Hilbert plane has a midpoint (that is, can be divided
into two congruent segments).
Problem 2. Prove that an isosceles triangle exists on any Hilbert plane. Give an example
of a Hilbert plane where equilateral triangles do not exist.
Problem 3 (Pons Asinorum). Prove that two base angles of an isosceles triangle are
equal. (The base angles are the angles opposite the equal sides.)
Problem 4. The lengths of four sides and a diagonal of a quadrilateral form the set
{1,2,2.8,5,7.5}. Find the length of the diagonal.
Problem 5. For any two distinct points A and C on a Hilbert plane, there exists at least
one point B such that B is between A and C.
Bonus problem 1. Add the Pythagorean theorem to Hilbert axioms I − III as an extra
axiom. Using the resulting system of axioms prove that there exists a triangle whose
angle sum is equal to 180◦ .
Non-Euclidean Geometry, spring term 2017
Problem solving session 1.
Faculty of Mathematics, NRU HSE, and Math in Moscow, IUM
Problem 1. Define segments and rays in a Hilbert plane.
Problem 2. Two distinct lines in a Hilbert plane can have at most one point in common.
Problem 3. Give an example of a Hilbert plane where equilateral triangles do not exist.
Problem 4. Any three distinct points on the same line can be labeled by A, B and C so
that B is between A and C.
Problem 5 (Postulate 4). Any two right angles are congruent to each other.
Problem 6 (Triangle inequality). (a) For any three points A, B and C in a Hilbert plane,
prove that
AB + BC ≥ AC
(b)The lengths of four sides and a diagonal of a quadrilateral form the set {3,5,7,11,17}.
Find the length of the diagonal.
Problem 7 (Discarded axiom of order). Any four distinct points on the same line can be
labeled by A, B, C, D so that B is between A and C, and also between A and D, while
C is between B and D, and also between A and D.
Problem 8 (Discarded axiom of congruence). For any three angles α, β and γ, if α = β
and α = γ, then β = γ.
Problem 9. Add to Hilbert axioms I − III as an extra axiom the following statement.
The angle sum of a triangle is strictly smaller than 180◦ . Using the resulting system
of axioms prove that for any acute angle there exists a point on its side such that the
perpendicular raised from this point is parallel to the other side of the angle.
Problem 10. Add to Hilbert axioms I − III as an extra axiom the following statement.
The angle sum of a triangle is strictly smaller than 180◦ . Using the resulting system of
axioms prove that there exist three points that lie neither on the same line nor on the
same circle.
Axioms of a Hilbert plane
Die Annahme dieser Eigenschaft der Linie ist
nichts als ein Axiom, durch welches wir erst
der Linie ihre Stetigkeit zuerkennen, durch
welches wir die Stetigkeit in die Linie
hineindenken.
Richard Dedekind
A Hilbert plane is a collection of points and lines together with incidence, betweenness and congruence relations that satisfy all axioms I − III listed below. The axioms
describe properties of undefined objects (points and lines) and relations between them
(incidence, betweenness, congruence, also described by verbs contain, lie between, and be
equal, respectively, and their synonyms).
Axioms of incidence geometry:
I1−2 For any two distinct points A and B, there exists a unique line containing A and B.
I3 Every line contains at least two points. There exist three noncollinear points (that is,
three points not all contained in a single line).
Axioms of order:
II1 If B is between A and C, then A, B and C are three distinct points on a line, and
also B is between C and A.
II2 For any two distinct points A and B, there exists a point C such that B is between
A and C.
II3 Given three distinct points on a line, at most one of them is between the other two.
II4 (Pasch) Let A, B, C be three noncollinear points, and let l be a line not containing
any of A, B, C. If l contains a point D lying between A and B, then it must also
contain either a point lying between A and C or a point lying between B and C.
Axioms of congruence: Congruence relation will be denoted by ≃. Following Hilbert,
by angle we mean the union of two rays originating at the same point.
III1 Given a line segment AB, and given a ray r originating at a point C, there exists a
unique point D on the ray r such that AB ≃ CD.
III2 If AB ≃ A′ B ′ and A′ B ′ ≃ A′′ B ′′ , then AB ≃ A′′ B ′′ (if two segments are each
congruent to a third segment, then they are congruent to each other).
III3 (Addition of segments) Let AB and BC be two segments on a line, and let A′ B ′ and
B ′ C ′ be two segments on another (or the same) line such that B is between A and C,
and B ′ is between A′ and C ′ If AB ≃ A′ B ′ and BC ≃ B ′ C ′ , then AC is congruent
to A′ C ′ .
III4 Given an angle ∠BAC and given a ray DF originating at a point D, there exists a
unique ray DE on a given side of the line DF such that ∠EDF ≃ ∠BAC.
III5 (SAS) Given triangles ABC and A′ B ′ C ′ , if AB ≃ A′ B ′ , AC ≃ A′ C ′ , and ∠BAC ≃
∠B ′ A′ C ′ , then ∆ABC ≃ ∆A′ B ′ C ′ .