0%

# Problem 562

Maximal perimeter

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) ~ 97.26729 and T(100) ~ 9157.64707.

• 顶点A，B和C都是在以原点为圆心，r为半径的圆内或圆周上的格点；
• 三角形内或边上没有其它的格点；
• 周长最大。