Thales' Theorem, Right Triangles + Napier's Rules - Universal Hyperbolic Geometry

Explore key concepts in universal hyperbolic geometry through this comprehensive 43-minute video lecture. Delve into the properties of right triangles, including singly, doubly, and triply right configurations. Learn about Thales' theorem and its relationship to Euclidean right triangles, with a focus on the spread as a crucial ratio. Discover the significance of quadrea in triangle analysis and master Napier's rules for solving right triangles. Follow along with proofs, engage in suggested exercises, and gain a deeper understanding of how to approach problems by creating right triangles. This lecture provides a solid foundation in advanced geometric concepts, blending theoretical knowledge with practical problem-solving techniques.

CONTENT SUMMARY: pg 1: @00:11 Review of the 4 main laws of trigonometry; pg 2: @ right triangles described; singly right, doubly right, triply right
pg 3: @06:16 Thales theorem; A kind of 'similarity' for right triangles @ ; In
classical hyperbolic geometry this result is obscured @
pg 4: @12:36 Thales theorem relationship to Euclidean RT; the spread as a crucial ratio @
pg 5: @17:41 Quadrea as the single most important number associated to a triangle; reminder on how to obtain an altitude @18:41 ; Quadrea are @
pg 6: @22:42 Napier's rules; suggested exercise @
pg 7: @ proof of Napier's rules;
pg 8: @30:02 proof continued; suggested algebra exercise @
pg 9: @36:49 When in doubt create some right triangles; exercise 29.1 @;
pg 10: @ exercise 29.2
pg 11: @ exercise 29.3,4