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December 4th, 2016, 02:28 PM   #1
Joined: Dec 2016
From: United States

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Proving perpendicular lines in hyperbolic geometry

Hi everyone,

This is actually a college level geometry question, I just wasn't sure where to post it. This question pertains to the hyperbolic plane.
Suppose ABO is an asymptotic triangle and <A = <B. If M is the midpoint of the finite side AB, prove that the line MO is perpendicular to AB.

Since O is an ideal point (on fundamental circle), we know that AO and BO are parallel lines such that they do not intersect. We also know that the degree of O is zero. Construct M on AB. Then, MO is also parallel to AO and BO.
Also, <OAM and <OBM are supplementary lines forming 180 degrees.

I'm stuck up to this point. I'm not sure how to prove that the angle formed at M is 90 and is perpendicular to AB. Could I go about this by saying that since AO and BO are parallel, then AB acts as a transversal? Then <A + <B = 180 since they are supplementary. Then, <A=<M since they are corresponding angles. and <M=<B since <A=<B.

Thanks for any help!
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