a) Make it as easy as possible, as you will explain to a novice!
b) Add figures when you explain the solutions.
c) Please add which theory you used to solve the questions
Qestions:
1. The corners of the triangle ∆P QR lie on the sides of the triangle ∆ABC (P on BC, Q on AC and R on AB). The points are located in such a way that AP, BQ and CR intersect at one point. Form the intersections U = AB · P Q, V = AC · P R and W = BC · QR. Show with a figure that U, V and W are on a straight line. Explain why this is so. You can use Desargue’s theorem which we may take as an axiom.
2. Formulate the dual for Pappus’ theorem and illustrate it in a figure. (Choose your lines carefully if you want the figure to fit on an A4!). Points must thus be replaced throughout the text by lines and vice versa. As well as lying on a line are replaced with intersecting at one point and vice versa.
3. Five different points A, B, C, A1, B1 lie on a conical section. Use Pascal’s theorem to obtain a sixth point (any) C1 on the cone section. You must therefore construct a sixth point on the ellipse (if it was an ellipse on which the five points were). Clue: send out a beam from A. C1 must lie on this beam. You can also select a parabola or hyperbola as a conic section if you want. If you use GeoGebra, you can construct an ellipse and place the points on it, and you will also have a result! See Figure 1.
(For the last two tasks, you may need to take a look at Pappus’ theorem. We concentrate on projective images from point to point. When an artist depicts something in nature on his canvas, he makes a perspective image. The artist’s eye is the center of the image. If a second artist depicts the painting on another painting, it is also a perspective depiction. Together, the two images form a projective image, from the object in nature to the other painting)
4. Perspective images can be combined into projective images. The points A, B, C on the line p shall be mapped on A1, B1, C1 on the line q by means of two perspective images so that A → A1, B → B1, C → C1, see figure 2. Use first A1 as center and then A. In the intermediate step, A, B, C must be plotted on a line r. Find that line. What happens if you instead use first B1 as the center and then B, it becomes the same intermediate line?
Figure : Projective image from line p to line q
5. Three points A, B, C on a line p must be imaged on the same line p in such a way that A → B, B → C, C → A. Then the projective image must consist of three perspective images. Make such a construction. Thus, three perspective centers (O1, O2, O3) and two extra lines (l1, l2) are needed. O1 depicts p → l1. O2 depicts l1 → l2 and O3 depicts l2 → p For which point X on p does X → ∞ apply?
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