Projective Plane: Axiomatic Approach A projective plane II is a set of points and subsets called lines that satisfy the following four axioms: • P1 Any two distinct points lie on a unique line. • P2 Any two lines meet in at least one point. • P3 Every line contains at least three points. • P4 There exists three non-collinear points. A projective plane is called finite, if there are only finitely many points on the plane. 1. Show that for any line 1 of II, there exists a point A s.t. A does not lie on 1. 2. Show that for any two distinct points A, B of II, there exits a line / which does not pass through neither of them. 3. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. 4. Show that for any given finite projective plane, there exists some n ≥ 2 s.t. every point in II lie on n + 1 lines and every line in II contains n+1 points. 5. Prove the duality principle we explained in lecture. 6. Use the duality principle to claim the dual statements of subquestions 1-4.

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Project: Thales and determining the height of an Egyptian Pyramid Published Edit ⠀ The Greek Mathematician, Thales of Miletus (c. 548/545 BCE) was said to have determined the height of the pyramids of Egypt using just geometry. Tests your skills to see if you can do the same. Below is a of an upright obelisk 20 meters in height, whose shadow as cast by the sun extends 27 meters to the left. The pyramid also casts a shadow that extends 198.45 meters to the left from the center of the pyramid. 27 meters 20 meters 198.45 meters X Answer the following questions by uploading an PDF document, or turn your work in on class on the date due (late class- room submissions are not allowed). (A) What method or relationship between geometric figures might Thales have used to determine the height of the pyramid? (B) What is the height of the pyramid according to the figure above? (include the correct unit of measurement in your answer)

Illuminate ← → × 0 | Geometry - Circle Assessment Question 2 Tools - Question 2 Use the diagram to solve for x. O ©2024 +45 A B C D 55° 90 1 0 P+ 45 Illuminate Education TM, Inc. Pause Q Zoom 145°