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## Duality in Projective Geometry and Spherical Geometry Problems

### Question Description

a) Make it as easy as possible, as you will explain to a novice!

b) Add to figures when you explain the solutions.

——

1. What is meant by ….

(a) ellipse, hyperbola and parabola;

https://1drv.ms/b/s!AgH2abs8e-hEgdxR1o9g9FaUWqspnQ

(b) spherical geometry,

https://1drv.ms/b/s!AgH2abs8e-hEgdxSiXuxBvKoXNz9rQ

(c) duality in projective geometry,

https://1drv.ms/b/s!AgH2abs8e-hEgdxTUcVPPCa7hcJ2fw

(d) hyperbolic geometry,

https://1drv.ms/b/s!AgH2abs8e-hEgdxUU5cIEkCfAWg0VA

(e) and fractals?

https://1drv.ms/b/s!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.

https://1drv.ms/b/s!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.

https://1drv.ms/b/s!AgH2abs8e-hEgdxXRWSdIYTKnPoh4w

(c) Construct √5

https://1drv.ms/b/s!AgH2abs8e-hEgdxXRWSdIYTKnPoh4w

—–

3. Some proofs:

(a) Prove Euclid’s 32nd theorem: the sum of the angles in a triangle is 180 degrees.

https://1drv.ms/b/s!AgH2abs8e-hEgdxYfoQ4KiA3xRFYqA

(b) Prove that in a circle the midpoint angle, ∠AMB, is twice the peripheral angle, ∠AP B.

https://1drv.ms/b/s!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.

https://1drv.ms/b/s!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)

https://1drv.ms/b/s!AgH2abs8e-hEgdxbY27C4ShBTriW1g

(e) Prove that the three medians of a triangle intersect at a point.

https://1drv.ms/b/s!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.

https://1drv.ms/b/s!AgH2abs8e-hEgdxdRPCh17nPOGieGQ

(g) Prove that the three bisectrices of a triangle intersect at a point.

https://1drv.ms/b/s!AgH2abs8e-hEgdxcwtnFSB_fCN4uAA

(h) Prove the bisector theorem for an inner angle.

https://1drv.ms/b/s!AgH2abs8e-hEgdxdRPCh17nPOGieGQ

(i) Prove Pythagoras’ theorem. You must know two different proofs. Euclid’s proof and another that you choose yourself.

https://1drv.ms/b/s!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: https://ibb.co/F3X78Vv

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