```From:           randall@ncr-sd.SanDiego.NCR.COM (Randall Rathbun)
Newsgroups:     sci.math
Subject:        Perfect Rational Triangle
Keywords:       rational triangle, rational sides,medians,area
Date:           13 Mar 87 22:12:31 GMT
Reply-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.math
Subject:        Re: Can anyone solve this puzzle
Date:           13 Dec 1993 17:39:06 -0600
Organization:   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

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 +0000
From:           Randall Rathbun <Randall_Rathbun@rc.trw.com>
Organization:   TRW ASD
To:             eppstein@ics.uci.edu
Subject:        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 +0000
From:           Randall Rathbun <Randall_Rathbun@rc.trw.com>
Organization:   TRW ASD
To:             eppstein@ics.uci.edu
Subject:        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 GMT
Newsgroups:     sci.math
Subject:        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 GMT
Newsgroups:     sci.math
Subject:        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.
```