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a) Make it as easy as possible, as you will explain to a novice! b) Add to figures when you explain the solutions. c) Please add which theory you used to solve the questions—— 1. What is meant by …. (a) ellipse, hyperbola and parabola;!AgH2abs8e-hEgdxR1o9g9FaUWqspnQ (b) spherical geometry,!AgH2abs8e-hEgdxSiXuxBvKoXNz9rQ (c) duality in projective geometry,!AgH2abs8e-hEgdxTUcVPPCa7hcJ2fw (d) hyperbolic geometry,!AgH2abs8e-hEgdxUU5cIEkCfAWg0VA (e) and fractals?!AgH2abs8e-hEgdxVZTT2qvAIssucYw—————– 2. Make below constructions with compasses and (ungraded) ruler. (a) Show how to divide a given distance into three equal parts. Enter the theorems you use.!AgH2abs8e-hEgdxWsuJZ4Nes2W7BKg (b) Show how to invert a distance OA. You must therefore master a procedure such as gives a point A1 such that OA OA1 = 1. A circle with radius 1 should be used.!AgH2abs8e-hEgdxXRWSdIYTKnPoh4w (c) Construct √5!AgH2abs8e-hEgdxXRWSdIYTKnPoh4w—–3. Some proofs: (a) Prove Euclid’s 32nd theorem: the sum of the angles in a triangle is 180 degrees.!AgH2abs8e-hEgdxYfoQ4KiA3xRFYqA (b) Prove that in a circle the midpoint angle, ∠AMB, is twice the peripheral angle, ∠AP B.!AgH2abs8e-hEgdxZ-K-ZwaORDdqnVA (c) A circle passes through all the corners of a quadrilateral. Show that the sum of two opposite angles is always equal to 180 degrees. And vice versa: In a quadrilateral, the sum of opposite angles is 180 degrees. Show that there is a circle that goes through all the corners of the quadrilateral.!AgH2abs8e-hEgdxakonb36BBHY4_jw (d) Prove the chord theorem both when the point P lies inside the circle and outside and touches it. (When the point P is outside the circle, the theorem is usually called the secant theorem)!AgH2abs8e-hEgdxbY27C4ShBTriW1g (e) Prove that the three medians of a triangle intersect at a point.!AgH2abs8e-hEgdxcwtnFSB_fCN4uAA (f) Prove that a point P lies on a bisector to the angle ∠BAC ⇔ The heights from P to the angle legs AB and AC are equal in length.!AgH2abs8e-hEgdxdRPCh17nPOGieGQ (g) Prove that the three bisectrices of a triangle intersect at a point.!AgH2abs8e-hEgdxcwtnFSB_fCN4uAA (h) Prove the bisector theorem for an inner angle.!AgH2abs8e-hEgdxdRPCh17nPOGieGQ (i) Prove Pythagoras’ theorem. You must know two different proofs. Euclid’s proof and another that you choose yourself.!AgH2abs8e-hEgdxfnTX_5ZcYFVPK7w (j) Derive the expression for the area of a spherical triangle, see the course book. I have posted a Youtube clip under the course documents about this. Feel free to make your own sketch of a ball or orange. course book:


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