tag:blogger.com,1999:blog-6555947.post7662271380062681129..comments2024-07-17T11:26:24.567-06:00Comments on The Geomblog: A pentagon problemSuresh Venkatasubramanianhttp://www.blogger.com/profile/15898357513326041822noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6555947.post-35374033855086651792007-02-01T14:18:00.000-07:002007-02-01T14:18:00.000-07:00Also, I have a slightly simpler (non-trigonometric...Also, I have a slightly simpler (non-trigonometric) proof of the impossibility of a regular pentagon with integer coordinates.<br /><br />Assume the contrary, and take the smallest such pentagon with one vertex at the origin. The clockwise-adjacent vertex is without loss of generality either (odd,odd), (odd,even), or (even,even).<br /><br />The (odd,odd) case corresponds to a side length whose square is 2 mod 4. The next clockwise-vertex must then necessarily be (even,even). Following parities around the pentagon leads to a contradiction.<br /><br />A similar argument rules out the (odd,even) case. The (even,even) case yields all even coordinates, which leads to a violation of the pentagon's minimality.<br /><br />This proof works for any regular n-gon with n odd.Davidhttps://www.blogger.com/profile/01360134038091251532noreply@blogger.comtag:blogger.com,1999:blog-6555947.post-91214385231303555372007-02-01T11:36:00.000-07:002007-02-01T11:36:00.000-07:00I can answer this because I once solved a somewhat...I can answer this because I once solved a somewhat related puzzle: find the smallest hexagon with integer sides/diagonals which can be inscribed in a circle. <br /><br />The answer to that puzzle is the inscribed hexagon of side lengths 3-5-3-5-3-5 (yielding diagonals of length 7 and 8). Picking 5 of the hexagon's vertices yields the desired pentagon. <br /><br />I discovered the hexagon by computer program (by enumerating triangles per circumradius and applying Ptolemy's Theorem).Davidhttps://www.blogger.com/profile/01360134038091251532noreply@blogger.comtag:blogger.com,1999:blog-6555947.post-13675545252386452942007-02-01T00:59:00.000-07:002007-02-01T00:59:00.000-07:00There are some answers in http://www.mathematik.tu...There are some answers in http://www.mathematik.tu-bs.de/preprints/199710.ps<br /><br />Or, search for "rational distances".Anonymousnoreply@blogger.com