Show that if the Pythagorean equation holds for all right tri- angles and if XC is...

 

Transcribed Image Text:

Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB = 2IJ can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) D E Figure 6.15 Saccheri quadrilateral associated to AABC. Saccheri triangle plane, that AD= CF = BE, hence that EDAB is a s quadrilateral with base ED, summit AB. Show that a r and its associated Saccheri quadrilateral have equal content- polygonal pieces and then reassemble these pieces to construct i.e., that you can dissect the Saccheri quadrilateral region into the triangular region. ← L Prove that the perpendicular bisector of AB is also perpendi Hence if the plane is hyperbolic, II is divergently parallel to cular to J. (Hint: Use a result about Saccheri quadrilaterals.) AB. Assume now the plane is real, so lengths can be assigned Prove that ED 21J. Deduce that AB > 21J (respectively AB = (Theorem 4.3) and the Saccheri-Legendre theorem applies. 2U) if the plane is hyperbolic (respectively is Euclidean). (d) Prove that K, F, and C are collinear if and only if AC BC (isosceles triangle). If that is the case, prove that F is the mid- point of CK iff the plane is Euclidean. If K, F, and C are not collinear and the plane is not Euclidean, prove that CF is not perpendicular to AB (ray CF does intersect AB at some point G in the case shown, where the angles at A and B are acute, by the crossbar theorem, but CG is not an altitude of the tri- angle if the plane is not Euclidean). 20 Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB 21J can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) = The remaining exercises are in hyperbolic geometry. You can use the results proved in this chapter as well as any results proved in neutral geometry in previous chapters. Do not use any of the Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB = 2IJ can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) D E Figure 6.15 Saccheri quadrilateral associated to AABC. Saccheri triangle plane, that AD= CF = BE, hence that EDAB is a s quadrilateral with base ED, summit AB. Show that a r and its associated Saccheri quadrilateral have equal content- polygonal pieces and then reassemble these pieces to construct i.e., that you can dissect the Saccheri quadrilateral region into the triangular region. ← L Prove that the perpendicular bisector of AB is also perpendi Hence if the plane is hyperbolic, II is divergently parallel to cular to J. (Hint: Use a result about Saccheri quadrilaterals.) AB. Assume now the plane is real, so lengths can be assigned Prove that ED 21J. Deduce that AB > 21J (respectively AB = (Theorem 4.3) and the Saccheri-Legendre theorem applies. 2U) if the plane is hyperbolic (respectively is Euclidean). (d) Prove that K, F, and C are collinear if and only if AC BC (isosceles triangle). If that is the case, prove that F is the mid- point of CK iff the plane is Euclidean. If K, F, and C are not collinear and the plane is not Euclidean, prove that CF is not perpendicular to AB (ray CF does intersect AB at some point G in the case shown, where the angles at A and B are acute, by the crossbar theorem, but CG is not an altitude of the tri- angle if the plane is not Euclidean). 20 Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB 21J can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) = The remaining exercises are in hyperbolic geometry. You can use the results proved in this chapter as well as any results proved in neutral geometry in previous chapters. Do not use any of the