From:randall@ncr-sd.SanDiego.NCR.COM (Randall Rathbun)Newsgroups:sci.mathSubject:Perfect Rational TriangleKeywords:rational triangle, rational sides,medians,areaDate:13 Mar 87 22:12:31 GMTReply-To:randall@ncr-sd.UUCP (0000-Randall Rathbun)Organization:NCR Corporation, Rancho Bernardo

In the interest of stating simple problems for the net... Does a perfect rational triangle exist? Question D21 in Richard K. Guy's Book "Unsolved Problems in Number Theory" (Springer-Verlag) rudimentarily discusses the problem. This triangle is defined as one with rational sides, rational medians, rational altitudes, and hence rational area. In my own research on a personal computer, the following 16 sets of almost perfect rational triangles were found where the 3 sides and medians are rational, but not the area. Rational Triangles with rational sides and medians Sides Medians Area 87 85 68 79 65.5 63.5 sqrt(7207200) 1 328 207 145 264.5 231.5 71 sqrt(105814800) 2 290 243 113 261.5 183.5 122 sqrt(179071200) 3 442 255 233 341.5 329.5 104 sqrt(521060400) 4 325 314 159 309.5 202 188.5 sqrt(602330400) 5 409 386 327 362.5 316 293.5 sqrt(3491888400) 6 477 446 277 440.5 320 284.5 sqrt(3670959600) 7 807 491 466 626 611.5 257.5 sqrt(10759694400) 8 907 774 581 791.5 656 512.5 sqrt(49744094400) 9 1664 1323 509 1481.5 1037.5 559 sqrt(77318060400) 10 1339 1099 810 1156 960.5 695.5 sqrt(197794674000) 11 1917 1306 877 1580.5 1340 564.5 sqrt(237944926800) 12 1929 1640 839 1740.5 1241 875.5 sqrt(466612146000) 13 2822 2007 1105 2385.5 1893.5 796 sqrt(769017916800) 14 1603 1524 1223 1439.5 1205 1125.5 sqrt(771033463200) 15 1973 1778 1401 1742.5 1462 1260.5 sqrt(1456477999200) 16 In addition, the following 2 sets of triangles were found that had 3 rational sides, 2 rational medians and a rational area. These appear to be much rarer than those above. Rational Triangles with 3 rational sides, 2 rational medians, and rational area Sides Medians Area 73 51 26 sqrt(3796) 48.5 17.5 420 875 626 291 sqrt(557580.25) 572 216.5 55440 I am satisfied that an infinite number of each type exist, although this is a strong conjecture. Also it should be noted that taking the medians, and using them as sides, leads to a conjugate triangle, so all these solutions listed come in pairs. In the second case, this means a rational triangle with 2 rational sides and 3 rational medians. Can anyone out there find more examples, especially of the second type listed, and mail the results back to me at randall@ncr-sd.UUCP? These will be forwarded to Prof. Richard K. Guy in Calgary, Alberta, of course, so he can append and update his information on Problem D21 in his excellent book. Thanks again, if your interest is stimulated by this article, and you decide to work upon finding more rational triangles. - Randall <randall@ncr-sd.UUCP>

From:gerry@macadam.mpce.mq.edu.au (Gerry Myerson)Newsgroups:sci.mathSubject:Re: Can anyone solve this puzzleDate:13 Dec 1993 17:39:06 -0600Organization:CeNTRe for Number Theory Research

In article <R.DELOACH.9.00100B58@larc.nasa.gov>, Dick DeLoach wrote: > > Given an equilateral triangle with an inclosed point, is there a > placement of the point such that the distance to each vertex is an integer > when the side of the triangle is also an integer? He's interested in a > general proof, not a numerical solution of a particular case. All help > gratefully received! This will be treated in the next edition of Richard Guy's book, Unsolved Problems in Number Theory. A draft I have reads, modulo some light editing, as follows: There are infinitely many solutions of the problem of integer distances from the corners of an equilateral triangle of side t. John Leech has sent us a neat and elementary proof of the fact that the points at rational distances from the vertices of any triangle with rational sides are dense in the plane of the triangle. This result was proved earlier by Almering. Arnfried Kemnitz notes that a = m^2 + n^2; b, c = m^2 +/- mn + n^2 with m = 2(u^2 -v^2), n = u^2 + 4uv + v^2 gives t = 8(u^2 - v^2)(u^2 + uv + v^2) and an infinity of solutions in which the points are neither on the sides nor the circumcircle of the triangle. A computer search showed that (57, 65, 73, 112) was the smallest such. Among the references Guy lists, the following may be relevant: J. H. J. Almering, Rational quadrilaterals, Indag Math 25 (1963) 192--199. As above, II, Indag Math 27 (1965) 290--304. T. G. Berry, Points at rational distances from the vertices of a triangle (no publication data given). T. G. Berry, Triangle distance problems and Kummer surfaces (no publication data given). Gerry Myerson Newsfeed unreliable. If you post a follow-up to this article, please send it to me by email, as well.

Date:Fri, 23 Aug 1996 19:22:59 +0000From:Randall Rathbun <Randall_Rathbun@rc.trw.com>Reply-To:Randall_Rathbun@rc.trw.comOrganization:TRW ASDTo:eppstein@ics.uci.eduSubject:perfect triangle

David: Recently Ralph Buchholz and I have proved that an infinity of Heron triangles with two rational medians do exist, using two Somos(5) series that Michael Somos discovered. We have also found two isolated Heron 2 med triangles that hint at more infinite chains, but despite our best efforts, we've not discovered their chains. Look for our paper in the American Mathematical Monthly. Recently, looking at some Legendre elliptic curves that Ralph and I discovered, and at some concordant forms of the 1/2 angle tangents of Heron 2 median triangles and 3 median triangles, I feel that the perfect triangle does NOT exist. The proof most likely will follow a Pocklington type of contradiction argument. - Randall Rathbun

Date:Fri, 23 Aug 1996 19:36:06 +0000From:Randall Rathbun <Randall_Rathbun@rc.trw.com>Reply-To:Randall_Rathbun@rc.trw.comOrganization:TRW ASDTo:eppstein@ics.uci.eduSubject:Two more interesting geometric problems

David: Two more problems that I love, but did not see on your web page, which, btw, is very nice, is the rational box, or properly called integer cuboid problem, whereby we look for a rectangular parallelopiped, such that its edges and diagonals and space diagonal are all integer. With some examination, it can be seen that 7 terms occur, and solutions have been found in 3 types, wherein the 7th parameter is irrational, the other 6 rational. Two types are the body and face cuboids, e.g. 104,153,672(697) face cuboid and 44,117,240(sqrt(73225)) body cuboid which are the smallest occurrances with integer sides. Edge cuboids exist also. I've found over 29,000 of the 3 types, they occur in a B:E:F 3:2:3 ratio. No perfect cuobids were found with the smallest edge < 1,281,000,000 This is problem D18 of Dr. Richard K. Guy's "Unsolved Problems in Number THeory" THe other problem is the nesting of right triangles inside each other, and how far chains of primitive right triangles can nest. I have found up to 16 such chains of right triangles. An example is (the pythagorean generators are listed) 36836,25135 31919,24920 31345,13312 24585,17576 23069,10780 18129,15140 23335,288 19177,5360 14957,7820 12033,9860 11455,6936 12425,1048 12253,220 8685,7864 8759,4738 (by way of explanation, if the pythagorean generator is 2,1 then the right triangle sides are 2*2*1, 2*2+1*1, and 2*2-1*1 which is the very familiar 3,4,5 right triangle) These chains are not easy to find. I hope you find these two problems interesting. - Randall Rathbun

From:jankok@cwi.nl (Jan Kok)Date:Thu, 19 Dec 1996 09:31:02 GMTNewsgroups:sci.mathSubject:Re: What is so special about 13-14-15 triangles?

In article <macleod-1812962123160001@ppp37.ravenet.com> macleod@sahara.wasteland.org (Mike Picollelli) writes: > >To all who can help me: > > > I was recently looking at a page full of math problems when I ran >across the words "famous 13-14-15 triangle" several times. I had never >even heard of this triangle before this, and had no idea that it was >considered famous. Can anyone clue me in as to why this triangle is so >special? Everyone I have asked has said that they have never even heard >of it. I appreciate all help on this subject. Apparently, it is not that famous, but it is special in the sense that if you consider all special lines, like bisector etc., some of them have rational lengths. Check for yourself. For example, one altitude is 12. -- --Jan Kok E-mail: Jan.Kok@cwi.nl | =#===-=========##= Address: CWI (dpt. NW) | -- ,___@ P.O. Box 94079 / 1090 GB Amsterdam | -- __/\ URL: http://www.cwi.nl/~jankok | ' /_

From:kfoster@rainbow.rmii.com (Kurt Foster)Date:21 Dec 1996 06:01:18 GMTNewsgroups:sci.mathSubject:Re: What is so special about 13-14-15 triangles?

Jan Kok (jankok@cwi.nl) wrote: : In article <macleod-1812962123160001@ppp37.ravenet.com> macleod@sahara.wasteland.org (Mike Picollelli) writes: : > : >To all who can help me: : > : > : > I was recently looking at a page full of math problems when I ran : >across the words "famous 13-14-15 triangle" several times. I had never : >even heard of this triangle before this, and had no idea that it was : >considered famous. Can anyone clue me in as to why this triangle is so : >special? Everyone I have asked has said that they have never even heard : >of it. I appreciate all help on this subject. : It's a "Heronian" or rational triangle. It's formed by taking a 9-12-15 right triangle (similar to the 3-4-5), and a 5-12-13 right triangle, and joining them along the side of length 12 to form a triangle. I guess it's the simplest rational triangle formed from two dissimilar integer-sided right triangles. There's a section in Maurice Kraitchik's "Mathematical Recreations" about integer-sided right triangles which mentions Heronian triangles.