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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.