Download Non-Euclidean Geometry, spring term 2017 Homework 4. Due date

Non-Euclidean Geometry, spring term 2017
Homework 4. Due date: March 6.
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. In the Euclidean plane, a rectangle ABCD has area 10. A point E lies on
the side CD. Find the area of a triangle ABE.
Problem 2. Let S be a unit sphere in R3 , and ∆ ⊂ S a triangle whose sides are arcs of
great circles. All angles of ∆ are equal to 90◦ . Find the area of ∆.
Problem 3. In a hyperbolic plane, there is a universal constant C such that the area of
any triangle is less than C.
Problem 4. (a) All vertices of a triangle ∆ ⊂ R2 in the Euclidean plane have integer
coordinates (i.e., belong to the lattice Z2 ⊂ R2 ), and ∆ does not contain any other lattice
points inside or at the boundary. Find the area of ∆.
(b) Give an example of a triangle from Problem 4 whose side lengths are greater than
1000.
Problem 5. In a hyperbolic plane, let ABCD be a Saccheri quadrilateral, that is, ∠A =
∠B = 90◦ and ∠C = ∠D. Show that the triangle ABC has smaller area than the triangle
ADC.
Bonus problem 4. Prove that in a hyperbolic Hilbert plane there exist three points that
lie neither on the same line nor on the same circle.
Non-Euclidean Geometry, spring term 2017
Problem solving session 4.
Faculty of Mathematics, NRU HSE, and Math in Moscow, IUM
Definition 1. By a hyperbolic plane we call a Hilbert plane that satisfies the following
axiom:
(L) Let l be any line and A a point not on l. Then there exist two rays AB and AC, not
lying on the same line, and not meeting l, such that any ray AE in the interior of the
angle BAC meets l.
We do not impose axioms V1−2 on a hyperbolic plane, however, you may use these axioms
if they make your solutions easier.
Problem 1. On the Euclidean plane, triangles on the same base, and with vertices lying
on a line parallel to the base, have the same area.
Problem 2. On a hyperbolic plane, triangles on the same base and with the same midline
have the same area.
Problem 3. Let S be a unit sphere in R3 , and ∆ ⊂ S a triangle whose sides are arcs of
great circles. The angles of ∆ are equal to α, β and γ. Find the area of ∆.
Problem 4. In a Hilbert plane H, let AD be a median of a triangle ABC. Suppose that
the angle ADB is acute. Compare the areas of the triangles ADB and ADC if
(a) H is Euclidean;
(b) H is hyperbolic.
Problem 5. (a) Prove that for any acute angle ∠AOB in a hyperbolic plane, there exists
a point E on the ray OA with the following property:
• If a point C is between O and E, then the perpendicular raised at C is parallel to
the line OB.
• If E is between O and a point C, then the perpendicular raised at C intersects
the line OB.
(b) Does the perpendicular raised at E intersect OB?
Problem 6 (Pick’s formula). Let ∆ ⊂ R2 be a lattice polygon in the Euclidean plane,
i.e., all vertices of ∆ have integer coordinates. Denote by P (∆) the number of lattice
points strictly inside ∆, and by E(∆) the number of lattice points at the boundary of ∆.
Prove that
1
area(∆) = P (∆) + E(∆) − 1.
2
Problem 7 (AAA). Prove or disprove the following statement for a hyperbolic Hilbert
plane. If three angles of one triangle are congruent to three angles of another triangle
then the triangles are congruent.
Problem 8. Show that on a hyperbolic plane there exists a pentagon with five direct
angles.