1969 Canadian MO Problems/Problem 9

Problem

Show that for any quadrilateral inscribed in a circle of radius $\displaystyle 1,$ the length of the shortest side is less than or equal to $\displaystyle \sqrt{2}$ .

Solution

Let $\displaystyle a,b,c,d$ be the edge-lengths and $\displaystyle e,f$ be the lengths of the diagonals of the quadrilateral. By Ptolemy's Theorem, $\displaystyle ab+cd = ef$ . However, each diagonal is a chord of the circle and so must be shorter than the diameter: $\displaystyle e,f \le 2$ and thus $\displaystyle ab+cd \le 4$ .

If $\displaystyle a,b,c,d > \sqrt{2}$ , then $\displaystyle ab+cd > 4,$ which is impossible. Thus, at least one of the sides must have length less than $\sqrt 2$ , so certainly the shortest side must.

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1969 Canadian MO Problems/Problem 9

Problem

Solution