1  Introduction

Let the base of the triangle have length b, and let a be the height of the altitude above the base. This is mentioned in section D18 of Richard Guy's book, see [guy].

PROBLEM: What integers n give base/height=n for integer-sided triangles?

The following discussion just provides the basic details of the analysis. Much more detailed accounts are available - PDF version or Postscript version.

Now, we have

a2  +  z2  =  c2

a2  +  ( b - z )2  =  d2

Now, if base/height = n, the second equation is

a2  +  ( z - n a )2  =  d2

For altitudes outside the triangle the equations are the same, except for z-na replaced by z+na. We thus consider the general system, with n positive or negative.

a2  +  z2  =  c2

a2  +  ( z - n a )2  =  d2

Clearly, we can assume that a and z have no common factors, so there exists integers p and q (of opposite parities) such that
(1) a = 2 p q  ,  z = p² - q²,
(2) a = p² - q² ,  z = 2 p q.

Assuming a = 2 p q and z = p² - q², then the equation for d is

d2  =  p4 - 4 n p3 q  +  ( 4 n2 + 2 ) p2 q2 +  4 n p q3  +  q4

This has an obvious rational point  ( p, q, d ) = ( 0, 1, 1 ), so is birationally equivalent to an elliptic curve. Using standard algebra, we can can link this equation to the curve

En  :  y2  =  x3  +  ( n2 + 2 ) x2  +  x

via the transformations p / q  =  ( n x + y ) / ( x + 1 ).

If, however, a = p² - q² and z = 2 p q, we have a different quartic for d², but leading to the same elliptic curve, with the relevant transformation

Thus the existence of solutions to the original problem is related to the rational points lying on the curve. There is the obvious point ( x, y ) = ( 0, 0 ), which gives p / q = 0 or p / q = -1, neither of which give non-trivial solutions. A little thought shows the points ( -1, ± n ), giving p / q = ?, p / q = 0 / 0, or p / q = 1, again failing to give non-trivial solutions.

It can be shown that the torsion subgroup must consist of the point at infinity,
( 0, 0 ), ( -1, ± n ). These points all lead to trivial solutions.

Thus, a non-trivial solution exists iff the rank of E_n is at least 1. If the rank is zero then no solution exists.
 

If we wish  (altitude / base) = m, then we can use the previous theory, with
n = 1 / m.

Defining v = m³ y,   u = m² x,   we get the system of elliptic curves F_m, given by

Fm : v2  =  u3 +  ( 2 m2 + 1 ) u2  +  m4 u

For  a = 2 p q and z = p² - q² , the relevant transformation is

As an example, for m = 13, the curve is v² =  u³  + 339 u²  +  28561u, which has the point  u = 19773/1156,  v = 30259281/39304.

The first transformation just given leads to p / q = 741/2278. This gives a = 3375996,  so that b = 259692, and x = -4640203. This gives c = 5738365,   d = 5950321.

The second transformation leads to the same triangle scaled by a factor of 2.

For both sets of curves, we used the Birch and Swinnerton-Dyer conjecture to estimate the rank for n and m in the range 2 to 999. The distribution of the ranks for E_n and for F_m are given in the following tables.
 
 

The solutions to the base/altitude problem can be found here, whilst the solutions for the altitude/base problem are here.

There are many values of N which have curves with rank 1 but no known generator. For the E_n curves, there are only 53 unsolved values of n ( as of May 2006), all with predicted heights over 110, with the maximum height estimated as 630. I have been really lucky with these curves as practically all problem values had a 2-isogenous curve with a point half the height, which was reasonably easy to find. As of August 2010, the number of unknowns is down to 13!

For F_m , however, the problem is significantly worse, with , currently, some 209 values of m unsolved. The 2-isogenous curve for these problems gives (usually) a height estimate of double the original curve. It is also the case that the heights of rank 1 generators are much larger for this family of curves. For example, the Birch and Swinnerton-Dyer conjecture predicts a height of 3580 for m = 919. I am gradually reducing these, but it takes time.