Given chords l and m of y that are not diameters Suppose the

Given chords l and m of y that are not diameters. Suppose the line extending m passes through the pole l. Prove that the line extending l passes through the pole of m. Please explain everything you do. Thank you!

an ideal an ideal n the direction of n. State the necessary Klein line and by a suffi- l and an n is uniquely determined r cient conditions and l in order and deter- mine Klein line. Translate this into a theorem in hyper bolic geometry K4. Given chords l m of y that are not diameters. Suppose extending m passes through the pole of l. Prove that the line extending l passes through the pole of m. (Hint: Use either equa- tion (2), in the last section of this chapter, or the theory of or- thogonal circles.) K-5. Use the Klein model to show that in the hyperbolic plane there ex- ists a pentagon with five right angles and there exists a hexagon with six right angles. (Hint: Begin with two lines having a com- mon perpendicular. Locate the poles of these two lines, then draw an appropriate line through each of the poles, etc) Does there ex- ist, for all n n-sided polygon n right angles? A\' of A A6. the following construction of the Klein reflection across m, which is simpler than the one in Figure 7.39. Let A be an end of m and let P be the pole of m. Join A to A and let s line cut y again at Join to P and let this line cut y at uith Ado\' (see Figure 7.46)

Solution

ANSWER:-

Proof:

We use the isomorphism between the Beltrami-Klein model and the Poincare disk model. The K-line l = A)(B from the Beltrami-Klein model is mapped to a P-line l\' in the Poincare disk model that meets in right angles. The P-line l\' is a portion of a circle, denoted in the Cartesian plane with center P(l). Likewise, the K-line m = P)(Q from the Beltrami-Klein model is mapped to a P-line m\' in the Poincare disk model that meets in right angles.

The circular arc m\' is a portion of a circle, denoted in the Cartesian plane with center P(m).The Corollary to Proposition tells us that meets in right angles if and only if inversion in maps onto itself. Notice how the roles of and can be interchanged in this statement.

If we assume that meets in right angles, then inversionin maps onto itself. Since l\' is a P-line, also meets in right angles,thus inversion in also maps to itself. It follows that the points P,Q of intersection between and are interchanged by the inversion in . This means that they are inverses under the inversion in . Points that are interchanged under the inversion lie on a diameter. Thus P,Q lie on a diameter of . In particular, the line Pbar Qbar is incident to P(l).

Interchanging the roles of and shows that the line Abar Bbar is incident to P(m).