We discuss Euclid's parallel postulate and the confusion it led to in the history of hyperbolic geometry. In Universal Hyperbolic Geometry we define the parallel to a line through a point, NOT the notion of parallel lines. This leads us to the useful construction of the double triangle of a triangle, and various perspective centers associated to it, the x, y and z points of a triangle. The x and z point lie on the ortho-axis, the y point generally does not.CONTENT SUMMARY: pg 1: @00:11 parallel's in hyperbolic geometrypg 2: @05:55 Better definitions? of parallel linespg 3: @09:29 Construction of the parallel P to L through a; no "P is parallel toL"pg 4: @13:01 Applying parallel's to a triangle; Double triangle in Euclidean geometry;pg 5: @14:51 Example of the double trilateral and double trianglepg 6: @16:33 Construction of? double triangle algebraically using st#1pg 7: @18:43 Double triangle midpoint theorem; Double triangle perspective theorem; The center of perspectivity x_point/double_point definedpg 8: @21:08 Exercise 18-1; x-point ortho-axis theorem; shxb cross-ratio theorem.pg 9: @22:42 Second double triangle perspective theorem;? y-point/second double point definedpg 10: @24:44 Double dual triangle perspective theorem; z-point revisited also called the double dual pointpg 11: @26:58 zbhs harmonic range theorem; zbxh harmonic range theorem; cg illustrations @28:15; UHG18 closing remarks @28:41 (THANKS to EmptySpaceEnterprise)
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