Problem Archive

Construct triangle $ABC$ such that:

• Vertices $A$, $B$ and $C$ are lattice points inside or on the circle of radius $r$ centered at the origin;
• the triangle contains no other lattice point inside or on its edges;
• the perimeter is maximum.

Let $R$ be the circumradius of triangle $ABC$ and $T(r) = R/r$.
For $r = 5$, one possible triangle has vertices $(-4,-3)$, $(4,2)$ and $(1,0)$ with perimeter $\sqrt{13}+\sqrt{34}+\sqrt{89}$ and circumradius $R = \sqrt {\frac {19669} 2 }$, so $T(5) = \sqrt {\frac {19669} {50} }$.
You are given $T(10) \approx 97.26729$ and $T(100) \approx 9157.64707$.

Find $T(10^7)$. Give your answer rounded to the nearest integer.