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Given a convex polygon. The circle is constructed for every triple of consecutive vertices of the polygon.We get the n circles. Select the circle with the largest radius. Prove that the circle contains the polygon.

My work so far:

$n=3 -$ triangle - obviously.

$n=4 -$

If $\angle B = \max \left\{A,B,C,D \right\}$ then $ABCD \in \omega_{ABC}$ enter image description here

$n\ge 5$. I need help here.

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    $\begingroup$ Perhaps induction on the number of sides? You can construct the three new circles associated with one new point and prove that either the old circle contains the new polygon, or one of the new ones does, and probably prove something about the radii at the same time. $\endgroup$ – abiessu Apr 14 '16 at 14:53

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