What Are the Connections Between Euclidean and Non-Euclidean Geometry?

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The discussion centers on a high school junior seeking guidance for a research project on hyperbolic and non-Euclidean geometry. The student aims to explore the connections between non-Euclidean and Euclidean geometry, as they feel more comfortable with the latter. They express a desire to avoid merely listing established facts and instead include personal analysis and insights. Suggestions include examining classical results in Euclidean geometry and their generalizations to non-Euclidean contexts, such as Napoleon's theorem. The student is looking for creative ways to approach their research question.
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research topic in non-euclid geo??--Help please!

hello! I am a high school junior trying to do an independent project in geometry and I've been reading about hyperbolic geometry--or non-euclid geo in general. i now have to pose a more specific research question within the topic and i really need some help!

i do not feel that i have enough knowledge/understanding to write a paper that is compeletely about hyperbolic geometry so i want to include some kindof relativity between non-euclid and euclid(which i have a better understanding of). i also find absolute(neutral) geometry interesting--just completely ignoring the troublesome parallel postulate!

my biggest problem is that as it IS a research paper, it can't just be a list of facts that other people have discovered a long time ago. I am not smart enough to come up w/ my own discovery or anything so I am trying to at least include my own analysis and viewpoint. (how else can i be creative?)
if you have any suggestions about the research question please please reply!
thank you!
 
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You could use classical results in Euclidean geometry and see how they generalize to non Euclidean geometries. E.g. Napoleon's theorem, or you can search the internet for "geometry + pdf" to find more examples.
 
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