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Constrained triangle (Posted on 2024-03-07) Difficulty: 3 of 5
Find a triangle with area 168, whose side lengths are integers, with all three vertices lying on a circle whose radius is a perfect square.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution solution | Comment 2 of 3 |
By Heron's formula,

A = sqrt(s(s-a)(s-b)(s-c))

and the radius of the circumcircle is

R = (abc) / sqrt((a + b + c)(b + c - a)(c + a - b)(a + b - c))

Trying all triangles with perimeters between 7 and 500:

for tot=7:500
  for a=1:tot/3
    for b=a:(tot-a)/2
      c=tot-a-b;
      s=(a+b+c)/2;
      area=sqrt(s*(s-a)*(s-b)*(s-c));
      if area == 168
        R=a*b*c/sqrt((a+b+c)*(b+c-a)*(b+a-c)*(a+c-b));
        sr=round(sqrt(R));
        if sr*sr==R
          disp([a b c R])
        end
      end
    end
  end
end

finds the sides of the triangle are

 14    30    40
 
and the radius of the circumcircle is 25.

  Posted by Charlie on 2024-03-07 09:37:02
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