= Quadratic path bound in a right triangle
{title2=$\sum_{e\in P}|e|^2\leq|a-b|^2$}
Given finitely many points in a <right triangle> with hypotenuse endpoints $a,b$, there is a path from $a$ to $b$ visiting them all with sum of squared edge lengths at most $\lVert a-b\rVert^2$. Repeated altitude subdivision reduces to cells containing one point. Joining the child paths preserves the squared-cost bound by the <Pythagorean theorem>; deleting an auxiliary right-angle vertex preserves it by the <law of cosines>.
Back to article page
