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