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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 discussesthe problem. This triangle is defined as one with rational sides, rationalmedians, rational altitudes, and hence rational area.In my own research on a personal computer, the following 16 sets of almostperfect rational triangles were found where the 3 sides and medians arerational, 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)      151973 1778 1401     1742.5 1462  1260.5     sqrt(1456477999200)     16 In addition, the following 2 sets of triangles were found that had 3 rationalsides, 2 rational medians and a rational area. These appear to be much rarerthan 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 astrong conjecture. Also it should be noted that taking the medians, and usingthem as sides, leads to a conjugate triangle, so all these solutions listedcome in pairs. In the second case, this means a rational triangle with 2rational 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 forwardedto Prof. Richard K. Guy in Calgary, Alberta, of course, so he can append andupdate his information on Problem D21 in his excellent book.Thanks again, if your interest is stimulated by this article, and you decideto 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 publicationdata given). Gerry MyersonNewsfeed 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 ofHeron triangles with two rational medians do exist, using twoSomos(5) series that Michael Somos discovered. We have also foundtwo isolated Heron 2 med triangles that hint at more infinitechains, but despite our best efforts, we've not discovered theirchains. Look for our paper in the American Mathematical Monthly.Recently, looking at some Legendre elliptic curves that Ralphand I discovered, and at some concordant forms of the 1/2 angletangents of Heron 2 median triangles and 3 median triangles, Ifeel that the perfect triangle does NOT exist. The proof mostlikely 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 calledinteger cuboid problem, whereby we look for a rectangularparallelopiped, such that its edges and diagonals and space diagonalare all integer. With some examination, it can be seen that 7terms occur, and solutions have been found in 3 types, whereinthe 7th parameter is irrational, the other 6 rational. Two typesare the body and face cuboids, e.g. 104,153,672(697) face cuboidand 44,117,240(sqrt(73225)) body cuboid which are the smallestoccurrances with integer sides. Edge cuboids exist also. I'vefound 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 < 1,281,000,000This is problem D18 of Dr. Richard K. Guy's "Unsolved Problemsin Number THeory"THe other problem is the nesting of right triangles inside eachother, and how far chains of primitive right triangles can nest.I have found up to 16 such chains of right triangles. An exampleis (the pythagorean generators are listed) 36836,2513531919,2492031345,1331224585,1757623069,1078018129,1514023335,28819177,536014957,782012033,986011455,693612425,104812253,2208685,78648759,4738(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 sensethat if you consider all special lines, like bisector etc.,some of them have rational lengths. Check for yourself. For example, onealtitude 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 9-12-15right triangle (similar to the 3-4-5), and a 5-12-13 right triangle, andjoining them along the side of length 12 to form a triangle.  I guess it'sthe simplest rational triangle formed from two dissimilar integer-sidedright triangles.  There's a section in Maurice Kraitchik's "Mathematical Recreations"about integer-sided right triangles which mentions Heronian triangles.