### UnivHypGeom18: Parallels and the double triangle

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)

Length:
29:35