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A few days ago, I drew attention to the question in which OP talked about the generation of triangles in a plane, for which the lengths of all sides, the area and radius of the inscribed circle are integers. In addition, all vertices must have different integer coordinates (6 different integers), the lengths of all sides are different and the triangles should not be rectangular. I prepared the answer to this question, but the question disappeared somewhere, so I designed my answer as a separate post.

The triangles in the plane, for which the lengths of all sides and the area  are integers, are called as Heronian triangles. See this very interesting article in the wiki about such triangles
https://en.wikipedia.org/wiki/Integer_triangle#Heronian_triangles

The procedure finds all triangles (with the fulfillment of all conditions above), for which the lengths of the two sides are in the range  N1 .. N2 . The left side of the range is an optional parameter (by default  N1=5). It is not recommended to take the length of the range more than 100, otherwise the operating time of the procedure will greatly increase. The procedure returns the list in which each triangle is represented by a list of  [list of coordinates of the vertices, area, radius of the inscribed circle, list of lengths of the sides]. Without loss of generality, one vertex coincides with the origin (obviously, by a shift it is easy to place it at any point). 

The procedure works as follows: one vertex at the origin, then the other two must lie on circles with integer and different radii  x^2+y^2=r^2. Using  isolve  command, we find all integer points on these circles, and then in the for loops we select the necessary triangles.


 

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restart;
HeronianTriangles:=proc(N2::posint,N1::posint:=5)
local k, r, S, L, Ch, Dist, IsOnline, c, P, p, A, B, C, a, b, s, ABC, cc, s1, T ;
uses combinat, geometry;
if N2<N1 then error "Should be N2>=N1" fi;
if N2<34 then return [] fi;
k:=0:
for r from max(N1,5) to N2 do
S:=[isolve(x^2+y^2=r^2)];
if nops(S)>4 then k:=k+1; L[k]:=select(s->s[1]<>0 and s[2]<>0,map(t->rhs~(convert(t,list)), S)); fi;
od:
L:=convert(L, list):
if type(L[1],symbol) then return [] fi;

Ch:=combinat:-choose([$1..nops(L)], 2):
Dist:=(A::list,B::list)->simplify(sqrt((A[1]-B[1])^2+(A[2]-B[2])^2));
IsOnline:=(A::list,B::list)->`if`(A[1]*B[2]-A[2]*B[1]=0, true, false);
k:=0:
for c in Ch do
for A in L[c[1]] do
for B in L[c[2]] do
if not IsOnline(A,B) and nops({A[],B[]})=4 then if type(Dist(A,B),posint) then
 k:=k+1; P[k]:=[A,B] fi; fi;
od: od: od:
P:=convert(P, list):
if type(P[1],symbol) then return [] fi;

k:=0:
for p in P do
point('A',0,0), point('B',p[1]), point('C',p[2]);
a:=simplify(distance('A','B')); b:=simplify(distance('A','C')); c:=simplify(distance('B','C'));
s:=sort([a,b,c]); s1:={a,b,c};
triangle(ABC,['A','B','C']);
incircle(cc,ABC);
r:=radius(cc);
if type(r,integer) and s[3]^2<>s[1]^2+s[2]^2 and nops(s1)=3 then k:=k+1; T[k]:=[[[0,0],p[]],area(ABC),r, [a,b,c]] fi;
od:
T:=convert(T,list);
if type(T[1],symbol) then return [] fi;
T;
end proc:

Examples of use of the procedure  HeronianTriangles

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T:=HeronianTriangles(100): # All the Geronian triangles, whose lengths of two sides do not exceed 100
nops(T);

256

 (1)
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Tp:=select(p->p[1,2,1]>0 and p[1,2,2]>0 and p[1,3,1]>0 and p[1,3,2]>0, T);

[[[[0, 0], [16, 30], [28, 21]], 252, 6, [34, 35, 15]], [[[0, 0], [30, 16], [21, 28]], 252, 6, [34, 35, 15]], [[[0, 0], [21, 28], [15, 36]], 168, 4, [35, 39, 10]], [[[0, 0], [28, 21], [36, 15]], 168, 4, [35, 39, 10]], [[[0, 0], [27, 36], [13, 84]], 900, 10, [45, 85, 50]], [[[0, 0], [36, 27], [84, 13]], 900, 10, [45, 85, 50]], [[[0, 0], [33, 44], [48, 36]], 462, 7, [55, 60, 17]], [[[0, 0], [44, 33], [36, 48]], 462, 7, [55, 60, 17]], [[[0, 0], [33, 44], [96, 28]], 1650, 15, [55, 100, 65]], [[[0, 0], [44, 33], [28, 96]], 1650, 15, [55, 100, 65]], [[[0, 0], [16, 63], [72, 21]], 2100, 20, [65, 75, 70]], [[[0, 0], [63, 16], [21, 72]], 2100, 20, [65, 75, 70]], [[[0, 0], [39, 52], [18, 80]], 1092, 12, [65, 82, 35]], [[[0, 0], [52, 39], [80, 18]], 1092, 12, [65, 82, 35]], [[[0, 0], [32, 60], [56, 42]], 1008, 12, [68, 70, 30]], [[[0, 0], [60, 32], [42, 56]], 1008, 12, [68, 70, 30]], [[[0, 0], [42, 56], [30, 72]], 672, 8, [70, 78, 20]], [[[0, 0], [56, 42], [72, 30]], 672, 8, [70, 78, 20]]]

 (2)
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Tr:=map(p->p+[2,1],Tp[1,1]);
with(geometry):
point(A,Tr[1]), point(B,Tr[2]), point(C,Tr[3]):
triangle(ABC,[A,B,C]):
simplify(distance(A,B)), simplify(distance(A,C)), simplify(distance(B,C));
local O:
incircle(c,ABC, centername=O):
draw([A,B,C, ABC, c(color=blue)], color=red, thickness=2, symbol=solidcircle, tickmarks = [spacing(1)$2], gridlines, scaling=constrained, view=[0..31,0..33], size=[800,550], printtext=true, font=[times, 18], axesfont=[times, 10]);

[[2, 1], [18, 31], [30, 22]]

  

34, 35, 15

  

 



Examples of triangles with longer sides

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T:=HeronianTriangles(1000,980):  # All the Geronian triangles, whose lengths of two sides lie in the range  980..1000
nops(T);

56

 (3)
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Tp:=select(p->p[1,2,1]>0 and p[1,2,2]>0 and p[1,3,1]>0 and p[1,3,2]>0, T);  # Triangles lying in the first quarter x>0, y>0
nops(%);

[[[[0, 0], [540, 819], [680, 714]], 85680, 80, [981, 986, 175]], [[[0, 0], [819, 540], [714, 680]], 85680, 80, [981, 986, 175]], [[[0, 0], [216, 960], [600, 800]], 201600, 168, [984, 1000, 416]], [[[0, 0], [960, 216], [800, 600]], 201600, 168, [984, 1000, 416]], [[[0, 0], [380, 912], [324, 945]], 31806, 31, [988, 999, 65]], [[[0, 0], [912, 380], [945, 324]], 31806, 31, [988, 999, 65]], [[[0, 0], [594, 792], [945, 324]], 277992, 216, [990, 999, 585]], [[[0, 0], [792, 594], [324, 945]], 277992, 216, [990, 999, 585]]]

  

8

 (4)
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Download Integer_Triangle1.mw

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geometry geronian-triangles

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