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;
(b) spherical geometry,
(c) duality in projective geometry,
(d) hyperbolic geometry,
(e) and fractals?
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.
(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.
(c) Construct √5
3. Some proofs:
(a) Prove Euclid’s 32nd theorem: the sum of the angles in a triangle is 180 degrees.
(b) Prove that in a circle the midpoint angle, ∠AMB, is twice the peripheral angle, ∠AP B.
(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.
(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)
(e) Prove that the three medians of a triangle intersect at a point.
(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.
(g) Prove that the three bisectrices of a triangle intersect at a point.
(h) Prove the bisector theorem for an inner angle.
(i) Prove Pythagoras’ theorem. You must know two different proofs. Euclid’s proof and another that you choose yourself.
(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: https://ibb.co/F3X78Vv
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