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UnivHypGeom10: Orthocenters exist!





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Uploaded on May 16, 2011

In classical hyperbolic geometry, orthocenters of triangles do not in general exist. Here in universal hyperbolic geometry, they do. This is a crucial building block for triangle geometry in this subject. The dual of an orthocenter is called an ortholine---also not seen in classical hyperbolic geometry!

This lecture also introduces a number of basic important definitions: that of side, vertex, couple, triangle, trilateral. We also introduce Desargues theorem and use it to define the polar of a point with respect to a triangle. The lecture culminates in the definition of the orthic line, orthostar and ortho-axis of a triangle. The ortho-axis will prove to be the most important line in hyperbolic triangle geometry.

CONTENT SUMMARY: pg 1: @00:15 Orthocenter of a triangle Basic definitions: a side, a vertex
pg 2: @03:49 more definitions: a couple, a triangle, a trilateral
pg 3: @06:37 A triangle has points, lines and vertices; a dual triangle;
pg 4: @10:45 A dual couple; Altitude line theorem;
pg 5: @12:52 Altitude point theorem;
pg 6: @15:02 Orthocenter theorem; examples of no orthocenter in Poincare model @17:37
pg 7: @18:24 Proof of Orthocenter theorem
pg 8: @29:43 The Ortholine theorem; the dual to the orthocenter theorem;examples from geometers sketchpad diagrams @31:40
pg 9: @32:28 Desargues theorem (a foundational theorem of projective geometry)
pg 10: @34:35 Establishing the polar of a point with respect to a triangle; cevian lines; the Desargue polar
pg 11: @39:07 Orthic axis, orthostar and ortho-axis; examples in Geometers Sketchpad @42:56 (THANKS to EmptySpaceEnterprise)

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