Is there a generalized Feuerbach point for an irregular non-Euclidean triangle? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:37:42Z http://mathoverflow.net/feeds/question/60968 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60968/is-there-a-generalized-feuerbach-point-for-an-irregular-non-euclidean-triangle Is there a generalized Feuerbach point for an irregular non-Euclidean triangle? Robert A. Russell 2011-04-07T16:41:30Z 2012-02-01T02:40:42Z <p>Is the circle externally tangent to the three excircles of an irregular non-Euclidean triangle internally tangent to the incircle of the triangle, the tangent point being a generalized Feuerbach point? In Euclidean plane geometry, the circle externally tangent to the excircles of an irregular triangle is internally tangent to the incircle at the Feuerbach point. My calculations indicate that this should similarly obtain for a non-Euclidean triangle, but I have found no proof. (It is possible that my calculations could, with difficulty, be turned into an analytic proof.) A circle is internally (externally) tangent to another circle if the distance between their centers is the difference (sum) of their radii. A triangle is irregular if it is not equilateral.</p> <p>In response to a helpful comment, my calculations indicate that the additional tangency does not obtain in higher dimensions, so I have revised the question to apply to only a two-dimensional space.</p> http://mathoverflow.net/questions/60968/is-there-a-generalized-feuerbach-point-for-an-irregular-non-euclidean-triangle/87195#87195 Answer by Robert A. Russell for Is there a generalized Feuerbach point for an irregular non-Euclidean triangle? Robert A. Russell 2012-02-01T02:40:42Z 2012-02-01T02:40:42Z <p>Akopyan pointed out to me that Hart proved this. Hart's 1861 article can be found at <a href="http://books.google.com/books?id=y9xEAAAAcAAJ&amp;pg=PA260#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.com/books?id=y9xEAAAAcAAJ&amp;pg=PA260#v=onepage&amp;q&amp;f=false</a>. Akopyan's article in translation appears at <a href="http://arxiv.org/pdf/1105.2153.pdf" rel="nofollow">http://arxiv.org/pdf/1105.2153.pdf</a>. My dynamic illustration appears at <a href="http://demonstrations.wolfram.com/NonEuclideanTriangleContinuum" rel="nofollow">http://demonstrations.wolfram.com/NonEuclideanTriangleContinuum</a>.</p>