1.Introduction

The problem of Pythagorean triangles with same area is mentioned in Guy's book[1],D21.
Is there an infinity of triples with same area?
Only 6 primitive triples are known.Primitive_Pythagorean_Triangles_with_Same_Area
It's known that there are 8 smallest Pythagorean triangles with all the same area below link. 
A055193-OEIS 
[6, 210, 840, 341880, 71831760, 64648584000, 2216650756320, 22861058133513600]
Above solutions are not primitive.

We show that there are Pythagorean triangles such that all triangles have same area for arbitrary n>1.
Furthermore, we construct 9 Pythagorean triangles with all the same area. 


2. Theorem

There are n Pythagorean triangles such that all triangles have same area for arbitrary n>1.


Proof.

A^2 + B^2 = C^2
A=k(x^2-y^2), B=2kxy, C=k(x^2+y^2), area=k^2xy(x^2-y^2).
m: square free number
We consider below equation.

mz^2=k^2xy(x^2-y^2)................................................(1)

Let X = mx/y, Y=m^2z/(ky^2) then we obtain elliptic curve below.

Y^2 = X^3-m^2X.....................................................(2)

This elliptic curve is related to congruent nunber problem.

If rank of elliptic curve (2) is geater than 0, then this elliptic curve has infinitely many rational points.

Thus we can obtain arbitrary n distinct rational solutions (A,B,C) by group law.

Area is given by lcm( x1y1(x1^2-y1^2), x2y2(x2^2-y2^2),....,xnyn(xn^2-yn^2)).
Hence we can construct the Pythagorean triangles such that all triangles have same area for arbitrary n>1.


Q.E.D.


3.Example

9 Pythagorean triangles with same area. 
Let m=1254.
9 rational solutions of mz^2=k^2xy(x^2-y^2):
[x,y,xy(x^2-y^2)]=[19, 8, 45144]
                  [49, 27, 2212056]
                  [539, 475, 16614998400]
                  [627, 49, 12004336344]
                  [147, 128, 98313600]
                  [25, 19, 125400]
                  [627, 529, 37575703704]
                  [11, 8, 5016]
                  [1862, 1859, 38640255654]
                  
area = lcm(45144, 2212056, 16614998400, 12004336344, 98313600, 125400, 37575703704, 5016, 38640255654)
     = 85065997314804105600                  
             
      A             B           C                area
[12892419540, 13196281280,  18448748500, 85065997314804105600]  
[10368506720, 16408533960,  19409943800, 85065997314804105600]
[ 4643503488, 36638713650,  36931794738, 85065997314804105600]
[32891483040,  5172524280,  33295715400, 85065997314804105600]
[ 4860237525, 35004872448,  35340670677, 85065997314804105600]
[ 6875957088, 24743027400,  25680657912, 85065997314804105600]
[ 5390243040, 31562954280,  32019912600, 85065997314804105600]
[ 7422908220, 22919856960,  24091895100, 85065997314804105600]
[ 523767960, 324823218720, 324823641000, 85065997314804105600]


4.Reference

[1] R. K. Guy, Unsolved Problems in Number Theory, 2nd edition, Springer-Verlag, 1994.